Contemporary Ideas on Ship Stability and Capsizing in Waves

Contemporary Ideas on Ship Stability and Capsizing in Waves

FLUID MECHANICS AND ITS APPLICATIONS Volume 96 Series Editor:

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Marcelo Almeida Santos Neves • Vadim L. Belenky Jean Otto de Kat • Kostas Spyrou • Naoya Umeda Editors

Contemporary Ideas on Ship Stability and Capsizing in Waves

Editors Marcelo Almeida Santos Neves Department of Naval Architecture and Oce Federal University of Rio de Janeiro Parque Tecnologico, Quadra 7 21941-972 Rio de Janeiro Brazil [email protected] Jean Otto de Kat A. P. Moeller - Maersk A/S Esplanaden 50 1098 Copenhagen Denmark [email protected] Naoya Umeda Department of Naval Architecture and Oce Graduate School of Eng Osaka University Yamadaoka Suita 2-1 565-0971 Osaka Japan [email protected]

Vadim L. Belenky Naval Surface Warfare Center Carderock D MacArthur Blvd 9500 20817-5700 West Bethesda Maryland USA [email protected] Kostas Spyrou School of Naval Architecture and Marine National Technical University of Athens Iroon Polytechnion, Zographou 9 157 73 Athens Greece [email protected]

Stability Criteria Evaluation and Performance Based Criteria Development for Damaged Naval Vessels, by Andrew James Peters © Copyright QinetiQ Limited 2010, by permission of QinetiQ Limited.

ISSN 0926-5112 ISBN 978-94-007-1481-6 e-ISBN 978-94-007-1482-3 DOI 10.1007/978-94-007-1482-3 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2011929340 © Springer Science+Business Media B.V. 2011 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Cover design: eStudio Calamar S.L. Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

Table of Contents Preface .................................................................................................................... xi List of Corresponding Authors ..........................................................................xiii

1 Stability Criteria Review of Available Methods for Application to Second Level Vulnerability Criteria ............................................................................................... 3 Christopher C. Bassler, Vadim Belenky, Gabriele Bulian, Alberto Francescutto, Kostas Spyrou and Naoya Umeda A Basis for Developing a Rational Alternative to the Weather Criterion: Problems and Capabilities ...................................................................................... 25 K.J. Spyrou Conceptualising Risk.............................................................................................. 47 Andrzej Jasionowski and Dracos Vassalos Evaluation of the Weather Criterion by Experiments and its Effect to the Design of a RoPax Ferry .............................................................................. 65 Shigesuke Ishida, Harukuni Taguchi and Hiroshi Sawada Evolution of Analysis and Standardization of Ship Stability: Problems and Perspectives ..................................................................................................... 79 Yury Nechaev SOLAS 2009 – Raising the Alarm ....................................................................... 103 Dracos Vassalos and Andrzej Jasionowski

2 Stability of the Intact Ship Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas ........................................................................................................ 119 A. Yucel Odabasi and Erdem Uçer Historical Roots of the Theory of Hydrostatic Stability of Ships ........................ 141 Horst Nowacki and Larrie D. Ferreiro The Effect of Coupled Heave/Heave Velocity or Sway/Sway Velocity Initial Conditions on Capsize Modeling .............................................................. 181 Leigh S. McCue and Armin W. Troesch

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The Use of Energy Build Up to Identify the Most Critical Heeling Axis Direction for Stability Calculations for Floating Offshore Structures ................. 193 Joost van Santen Some Remarks on Theoretical Modelling of Intact Stability............................... 217 N. Umeda, Y. Ohkura, S. Urano, M. Hori and H. Hashimoto

3 Parametric Rolling An Investigation of Head-Sea Parametric Rolling for Two Fishing Vessels.................................................................................................................. 231 Marcelo A.S. Neves, Nelson A. Pérez, Osvaldo M. Lorca and Claudio A. Rodríguez Simple Analytical Criteria for Parametric Rolling............................................... 253 K.J. Spyrou Experimental Study on Parametric Roll of a Post-Panamax Containership in Short-Crested Irregular Waves ................................................. 267 Hirotada Hashimoto, Naoya Umeda and Akihiko Matsuda Model Experiment on Parametric Rolling of a Post-Panamax Containership in Head Waves .............................................................................. 277 Harukuni Taguchi, Shigesuke Ishida, Hiroshi Sawada and Makiko Minami Numerical Procedures and Practical Experience of Assessment of Parametric Roll of Container Carriers ............................................................. 295 Vadim Belenky, Han-Chang Yu and Kenneth Weems Parametric Roll and Ship Design ......................................................................... 307 Marc Levadou and Riaan van’t Veer Parametric Roll Resonance of a Large Passenger Ship in Dead Ship Condition in All Heading Angles......................................................................... 331 Abdul Munif, Yoshiho Ikeda, Tomo Fujiwara and Toru Katayama Parametric Rolling of Ships – Then and Now...................................................... 347 J. Randolph Paulling

4 Broaching-to Parallels of Yaw and Roll Dynamics of Ships in Astern Seas and the Effect of Nonlinear Surging .................................................................... 363 K.J. Spyrou

Table of Contents vii

Model Experiment on Heel-Induced Hydrodynamic Forces in Waves for Realising Quantitative Prediction of Broaching............................. 379 H. Hashimoto, Naoya Umeda and Akihiko Matsuda Perceptions of Broaching-To: Discovering the Past ............................................ 399 K.J. Spyrou

5 Nonlinear Dynamics and Ship Capsizing Use of Lyapunov Exponents to Predict Chaotic Vessel Motions ........................ 415 Leigh S. McCue and Armin W. Troesch Applications of Finite-Time Lyapunov Exponents to the Study of Capsize in Beam Seas ...................................................................................... 433 Leigh S. McCue Nonlinear Dynamics on Parametric Rolling of Ships in Head Seas .................... 449 Marcelo A.S. Neves, Jerver E.M. Vivanco and Claudio A. Rodríguez

6 Roll Damping A Simple Prediction Formula of Roll Damping of Conventional Cargo Ships on the Basis of Ikeda’s Method and Its Limitation ......................... 465 Yuki Kawahara, Kazuya Maekawa and Yoshiho Ikeda A Study on the Characteristics of Roll Damping of Multi-Hull Vessels............. 487 Toru Katayama, Masanori Kotaki and Yoshiho Ikeda

7 Probabilistic Assessment of Ship Capsize Capsize Probability Analysis for a Small Container Vessel ................................ 501 E.F.G. van Daalen, J.J. Blok and H. Boonstra Efficient Probabilistic Assessment of Intact Stability.......................................... 515 N. Themelis and K.J. Spyrou Probability of Capsizing in Beam Seas with Piecewise Linear Stochastic GZ Curve ............................................................................................ 531 Vadim Belenky, Arthur M. Reed and Kenneth M. Weems Probabilistic Analysis of Roll Parametric Resonance in Head Seas.................... 555 Vadim L. Belenky, Kenneth M. Weems, Woei-Min Lin and J. Randolf Paulling

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8 Environmental Modeling Sea Spectra Revisited ........................................................................................... 573 Maciej Pawłowski On Self-Repeating Effect in Reconstruction of Irregular Waves......................... 589 Vadim Belenky New Approach To Wave Weather Scenarios Modeling ...................................... 599 Alexander B. Degtyarev

9 Damaged Ship Stability Effect of Decks on Survivability of Ro–Ro Vessels ............................................ 621 Maciej Pawłowski Experimental and Numerical Studies on Roll Motion of a Damaged Large Passenger Ship in Intermediate Stages of Flooding................................... 633 Yoshiho Ikeda, Shigesuke Ishida, Toru Katayama and Yuji Takeuchi Exploring the Influence of Different Arrangements of Semi-Watertight Spaces on Survivability of a Damaged Large Passenger Ship............................. 643 Riaan van’t Veer, William Peters, Anna-Lea Rimpela and Jan de Kat Time-Based Survival Criteria for Passenger Ro-Ro Vessels............................... 663 Andrzej Jasionowski, Dracos Vassalos and Luis Guarin Pressure-Correction Method and Its Applications for Time-Domain Flooding Simulation ............................................................................................. 689 Pekka Ruponen

10 CFD Applications to Ship Stability Applications of 3D Parallel SPH for Sloshing and Flooding............................... 709 Liang Shen and Dracos Vassalos Simulation of Wave Effect on Ship Hydrodynamics by RANSE........................ 723 Qiuxin Gao and Dracos Vassalos A Combined Experimental and SPH Approach to Sloshing and Ship Roll Motions......................................................................................................... 735 Luis Pérez-Rojas, Elkin Botía-Vera, José Luis Cercos-Pita, Antonio Souto-Iglesias, Gabriele Bulian and Louis Delorme

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11 Design for Safety Design for Safety with Minimum Life-Cycle Cost.............................................. 753 Romanas Puisa, Dracos Vassalos, Luis Guarin and George Mermiris

12 Stability of Naval Vessels Stability Criteria Evaluation and Performance Based Criteria Development for Damaged Naval Vessels........................................................... 773 Andrew J. Peters and David Wing A Naval Perspective on Ship Stability ................................................................. 793 Arthur M. Reed

13 Accident Investigations New Insights on the Sinking of MV Estonia........................................................ 827 Andrzej Jasionowski and Dracos Vassalos Experimental Investigation on Capsizing and Sinking of a Cruising Yacht in Wind ...................................................................................................... 841 Naoya Umeda, Masatoshi Hori, Kazunori Aoki, Toru Katayama and Yoshiho Ikeda Author Index....................................................................................................... 855

Preface During the last decade significant progress has been made in the field of ship stability: the understanding of ship dynamics and capsize mechanisms has increased through improved simulation tools and dedicated model tests, design regulations have been adapted based on better insights into the physics of intact and damaged vessels, and ships can be operated with better knowledge of the dynamic behaviour in a seaway. Yet in spite of the progress made, numerous scientific and practical challenges still exist with regard to the accurate prediction of extreme motion and capsize dynamics for intact and damaged vessels, the probabilistic nature of extreme events, criteria that properly reflect the physics and operational safety of an intact or damaged vessel, and ways to provide relevant information on safe ship handling to ship operators. This book provides a comprehensive review of the above issues through the selection of representative papers presented at the unique series of international workshops and conferences on ship stability held between 2000 and 2009. The first conference on the stability of ships and ocean vehicles was held in 1975 in Glasgow (STAB ‘75) and the first international workshop on ship stability was held in 1995 in Glasgow. The aim of the workshop was to address issues related to ship capsize and identify ways to accelerate relevant developments. Since then the workshops have been held each year in succession, whereby each third year a STAB conference has been held as a cap of the preceding workshops and to present the latest developments. The editorial committee has selected papers for this book from the following events: STAB 2000 Conference (Launceston, Tasmania), 5th Stability Workshop (Trieste, 2001), 6th Stability Workshop (Long Island, 2002), STAB 2003 Conference (Madrid), 7th Stability Workshop (Shanghai, 2004), 8th Stability Workshop (Istanbul, 2005), STAB 2006 Conference (Rio de Janeiro), 9th Stability Workshop (Hamburg, 2007), 10th Stability Workshop (Daejeon, 2008), and STAB 2009 Conference (St. Petersburg). The papers have been clustered around the following themes: Stability Criteria, Stability of the Intact Ship, Parametric Rolling, Broaching, Nonlinear Dynamics, Roll Damping, Probabilistic Assessment of Ship Capsize, Environmental Modelling, Damaged Ship Stability, CFD Applications, Design for Safety, Naval Vessels, and Accident Investigations. We would like to express our sincere gratitude to all of the authors who have presented papers at the conferences and workshops; to the organizers and all of the participants of these events; to the members of the International Standing Committee for the Stability of Ships and Ocean Vehicles; and to Professor Marcelo Neves for his role as main editor of this book. Dr. Jan Otto de Kat Chairman of the International Standing Committee for the Stability of Ships and Ocean Vehicles (on behalf of the editorial committee)

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List of Corresponding Authors 1 Stability Criteria Review of Available Methods for Application to Second Level Vulnerability Criteria Vadim Belenky [email protected] A Basis for Developing a Rational Alternative to the Weather Criterion: Problems and Capabilities K.J. Spyrou [email protected] Conceptualising Risk Dracos Vassalos [email protected] Evaluation of the Weather Criterion by Experiments and its Effect to the Design of a RoPax Ferry Shigesuke Ishida [email protected] Evolution of Analysis and Standardization of Ship Stability: Problems and Perspectives Yury Nechaev [email protected] SOLAS 2009 – Raising the Alarm Dracos Vassalos [email protected] 2 Stability of the Intact Ship Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas E. Uçer [email protected] Historical Roots of the Theory of Hydrostatic Stability of Ships Horst Nowacki [email protected]

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xiv List of Corresponding Authors

The Effect of Coupled Heave/Heave Velocity or Sway/Sway Velocity Initial Conditions on Capsize Modeling Leigh S. McCue [email protected] The Use of Energy Build Up to Identify the Most Critical Heeling Axis Direction for Stability Calculations for Floating Offshore Structures Joost van Santen [email protected] Some Remarks on Theoretical Modelling of Intact Stability Naoya Umeda [email protected] 3 Parametric Rolling An Investigation of Head-Sea Parametric Rolling for Two Fishing Vessels Marcelo de Almeida Santos Neves [email protected] Simple Analytical Criteria for Parametric Rolling K.J. Spyrou [email protected] Experimental Study on Parametric Roll of a Post-Panamax Containership in Short-Crested Irregular Waves Hirotada Hashimoto [email protected] Model Experiment on Parametric Rolling of a Post-Panamax Containership in Head Waves Harukuni Taguchi [email protected] Numerical Procedures and Practical Experience of Assessment of Parametric Roll of Container Carriers Vadim Belenky [email protected] Parametric Roll and Ship Design Riaan van’t Veer [email protected]

List of Corresponding Authors xv

Parametric Roll Resonance of a Large Passenger Ship in Dead Ship Condition in All Heading Angles Toru Katayama [email protected] Parametric Rolling of Ships – Then and Now J. Randolph Paulling [email protected] 4 Broaching-to Parallels of Yaw and Roll Dynamics of Ships in Astern Seas and the Effect of Nonlinear Surging K.J. Spyrou [email protected] Model Experiment on Heel-Induced Hydrodynamic Forces in Waves for Realising Quantitative Prediction of Broaching Hirotada Hashimoto [email protected] Perceptions of Broaching-To: Discovering the Past K.J. Spyrou [email protected] 5 Nonlinear Dynamics and Ship Capsizing Use of Lyapunov Exponents to Predict Chaotic Vessel Motions Leigh S. McCue [email protected] Applications of Finite-Time Lyapunov Exponents to the Study of Capsize in Beam Seas Leigh S. McCue [email protected] Nonlinear Dynamics on Parametric Rolling of Ships in Head Seas Marcelo A.S. Neves, [email protected] 6 Roll Damping A Simple Prediction Formula of Roll Damping of Conventional Cargo Ships on the Basis of Ikeda’s Method and Its Limitation Yoshiho Ikeda [email protected]

xvi List of Corresponding Authors

A Study on the Characteristics of Roll Damping of Multi-hull Vessels Toru Katayama [email protected] 7 Probabilistic Assessment of Ship Capsize Capsize Probability Analysis for a Small Container Vessel E.F.G. van Daalen [email protected] Efficient Probabilistic Assessment of Intact Stability K.J. Spyrou [email protected] Probability of Capsizing in Beam Seas with Piecewise Linear Stochastic GZ Curve Vadim Belenky [email protected] Probabilistic Analysis of Roll Parametric Resonance in Head Seas Vadim L. Belenky [email protected] 8 Environmental Modeling Sea Spectra Revisited Maciej Pawłowski [email protected] On Self-Repeating Effect in Reconstruction of Irregular Waves Vadim Belenky [email protected] New Approach To Wave Weather Scenarios Modeling Alexander B. Degtyarev [email protected] 9 Damaged Ship Stability Effect of Decks on Survivability of Ro–Ro Vessels Maciej Pawłowski [email protected]

List of Corresponding Authors xvii

Experimental and Numerical Studies on Roll Motion of a Damaged Large Passenger Ship in Intermediate Stages of Flooding Yoshiho Ikeda [email protected] Exploring the Influence of Different Arrangements of Semi-Watertight Spaces on Survivability of a Damaged Large Passenger Ship Riaan van’t Veer [email protected] Time-Based Survival Criteria for Passenger Ro-Ro Vessels Dracos Vassalos [email protected] Pressure-Correction Method and Its Applications for Time-Domain Flooding Simulation Pekka Ruponen [email protected] 10 CFD Applications to Ship Stability Applications of 3D Parallel SPH for Sloshing and Flooding Dracos Vassalos [email protected] Simulation of Wave Effect on Ship Hydrodynamics by RANSE Dracos Vassalos [email protected] A Combined Experimental and SPH Approach to Sloshing and Ship Roll Motions Luis Pérez-Rojas [email protected] 11 Design for Safety Design for Safety with Minimum Life-Cycle Cost Dracos Vassalos [email protected]

xviii List of Corresponding Authors

12 Stability of Naval Vessels Stability Criteria Evaluation and Performance Based Criteria Development for Damaged Naval Vessels Andrew J. Peters [email protected] A Naval Perspective on Ship Stability Arthur M. Reed [email protected] 13 Accident Investigations New Insights on the Sinking of MV Estonia Dracos Vassalos [email protected] Experimental Investigation on Capsizing and Sinking of a Cruising Yacht in Wind Naoya Umeda [email protected]

1 Stability Criteria

Review of Available Methods for Application to Second Level Vulnerability Criteria Christopher C. Bassler*, Vadim Belenky*, Gabriele Bulian**, Alberto Francescutto**, Kostas Spyrou***, Naoya Umeda**** *Naval Surface Warfare Center Carderock Division (NSWCCD) - David Taylor Model Basin, USA; **Trieste University; Italy; ***Technical University of Athens, Greece; ****Osaka University, Japan Abstract The International Maritime Organization (IMO) has begun work on the development of next generation intact stability criteria. These criteria are likely to consist of several levels: from simple to complex. The first levels are expected to contain vulnerability criteria and are generally intended to identify if a vessel is vulnerable to a particular mode of stability failure. These vulnerability criteria may consist of relatively simple formulations, which are expected to be quite conservative to compensate for their simplicity. This paper reviews methods which may be applicable to the second level of vulnerability assessment, when simple but physics-based approaches are used to assess the modes of stability failure, including pure-loss of stability, parametric roll, surf-riding, and dead-ship condition.

1 Introduction Current stability criteria have been used for several decades to determine the level of safety for both existing and novel ship designs. However, their current state is not representative of the level of our understanding about the mechanisms of dynamic stability failure of ships. Acknowledging this deficiency, the IMO has begun work on the development of next generation intact stability criteria to address problems related to dynamic phenomena and expand the applicability of criteria to current and future ship designs. The new generation criteria should include a range of fully dynamic aspects in their formulation. A thorough critical analysis of the existing situation at the beginning of the revision of the Intact Stability Code and a description of the present stateof-the-art can be found in a series of works by Francescutto (2004, 2007). The current framework of new generation intact stability criteria started to take shape at the 50th session of IMO Sub-committee on Stability and Load lines and M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_1, © Springer Science+Business Media B.V. 2011

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on Fishing Vessels Safety (SLF) with SLF-50/4/4 (2007), submitted by Japan, the Netherlands, and the United States and SLF-50/4/12 (2007), submitted by Italy. A significant contribution was also made in SLF-50.INF.2 (2007), submitted by Germany. These discussions resulted in a plan of development of new generation of criteria (see Annex 6 of SLF 50/WP.2). Following this plan, the intersessional Correspondence Group on Intact Stability developed the framework in a form of the working document that can be found in Annex 2 of SLF 51/4/1 (2008), submitted by Germany. Annex 3 of this paper contains the draft terminology agreed upon for the new criteria that was also developed by the Correspondence Group. The most current version of the framework document can be found in Annex 1 of the report of the Working Group on Intact Stability (SLF 51/WP.2). To facilitate these discussions, Belenky, et al. (2008) presented a broad review of issues related with development of new criteria, including the physical background, methodology, and available tools. It is the intention of the authors of this paper to provide a follow-up review, with a particular focus on methods for application to second level vulnerability criteria. The views and opinions expressed in this paper are solely and strictly those of the authors, mainly for facilitating wider and deeper discussion inside the research community. They do not necessarily reflect those of the delegations that involve the authors, the intersessional correspondence group on intact stability, or the working group on intact stability of the International Maritime Organization. The contents of this paper also do not necessarily indicate a consensus of opinion among the authors, but the authors agree on the need for further discussion of the contents.

2 Overview of New Generation of Intact Stability Criteria The major modes of stability failures were listed in section 1.2 of the 2008 IS Code, part A. They include:  Phenomena related to righting lever variations, such as parametric roll and pure loss of stability;  Resonant roll in dead ship condition defined by SOLAS regulation II-1/3  Broaching and manoeuvring-related problems in waves A multi-tier structure of new criteria has been identified from the ongoing discussion at IMO. Each of the four identified stability failure modes: pure-loss of stability, parametric roll, surf-riding and broaching, and dead ship conditions is evaluated using criteria with different levels of sophistication. The intention of the vulnerability criteria is to identify ships that may be vulnerable to dynamic stability failures, and are not sufficiently covered by existing regulation (SLF 51/WP.2, Annex 1).

Review of Available Methods for Application to Second Level Vulnerability Criteria

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A first level of criteria (vulnerability criteria) could be followed by other stability assessment methods with increasing levels of sophistication. The first level of the stability assessment is expected to be relatively simple, preferably geometry based if possible. The function of this first level is to distinguish conventional ships, which are obviously not vulnerable to stability failure. The second tier is expected to be more sophisticated and assess vulnerability to dynamic stability with enough confidence that the application of direct calculation could be justified. The expected complexity of the second level vulnerability criteria should consist of a spreadsheet-type calculation. An outline of the process is shown in Fig. 1. The methods will most likely be different for each stability failure mode. Ship Design

Stability Failure Mode

Vulnerability Criteria Level 1

For each mode

Fail Pass IMO IS Code

Performance-Based Criteria

Vulnerability Criteria Level 2

Fail Pass IMO IS Code

Fig. 1. The proposed assessment process for next generation intact stability criteria

3 Pure Loss of Stability Differences in the change of stability in waves, in comparison with calm water, were known to naval architects since late 1800s (Pollard & Dudebout, 1892; Krylov 1958). However, it was uncommon until the 1960s that attempts to calculate the change of stability in waves (Paulling, 1961) and evaluate it with a series of model tests (Nechaev, 1978; available in English from Belenky & Sevastianov, 2007) were made. As a distinct mode of stability failure, pure-loss of stability was identified during the model experiments in San-Francisco Bay (Paulling, et al., 1972, 1975; summary available from Kobylinski & Kastner, 2003). Accurate calculation of the change of stability in waves presents certain challenges, especially at high speed, as the pressure around the hull and waterplane shape are influenced by the nonlinear interaction between encountered, reflected, and radiated waves (Nechaev, 1989).

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The physical mechanism of pure-loss of stability is rather simple: if the stability is reduced for a sufficiently long time, a ship may capsize or attain large roll angle (see Fig, 2, taken from Belenky, et al., 2008).

200

Time history of roll and capsizing

Righting / Heeling Arm, m 1 Wave trough Magnitude of heeling Calm water moment 0.5 Wave crest

150 0 100 50

10 20 30 40 50

 deg 60

-0.5 Phase of heeling moment Phase of encounter wave

0

50

100

150

t, s

50

Fig. 2. Capsizing due to pure-loss of stability

This phenomenon is presently addressed, although in a very short and qualitative form, by the MSC.1/Circ.1228 (2007). MSC.1/Circ.1228 does not provide any ship-dependent information concerning the considered failure mode. The difficulty of developing criteria for pure-loss of stability is not limited to the calculation of the GZ curve in waves. A probabilistic characterization of the associated stability changes is also difficult, due to nonlinear nature of stability changes in waves. Boroday (1967, 1968) developed a method for the assessment of statistical characteristics for restoring moments in waves, followed by an energy balancebased method for evaluation of the probability of capsizing (Boroday & Netsevatev, 1969). Using energy balance methods for changing stability in waves was also the focus of Kuo & Vassalos (1986). Dunwoody (1989a, 1989b), Palmquist (1994), and Bulian & Francescutto (2006a) considered statistical characteristics of GM and other elements of the righting moment as a stochastic process. The alternative approach of the “effective wave” was proposed by Grim (1961). In this method, the length of the wave is equal to ship length— the wave is unmovable. However, its amplitude is a random process (Fig. 3).

Fig. 3. A concept of an effective wave

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The amplitude of the effective wave is calculated in order to minimize the difference between the effective and encountered waves. Umeda, et al. (1993) performed comparative calculations and suggested the effective wave to be of satisfactory accuracy for engineering calculations. The effective wave approach was used in a number of works; of special interest are those with a “regulatory perspective.” Helas (1982) discussed the reduction of the righting arm in waves. To consider the duration where stability is reduced, the amplitude of the effective wave was averaged over a time sufficient to roll to a large angle, about 60% of natural roll period. As a result, the average effective wave amplitude becomes a function of Froude number. Analyzing results of the calculation performed for different vessels in different sea conditions, recommendations were formulated for a stability check in waves. Small ships and ships with “reduced stability” were meant to be checked using the GZ curve— evaluated for the effective wave with averaged amplitude against conventional calm water stability criteria. Umeda and Yamakoshi, (1993) used the effective wave to evaluate the probability of capsizing due to pure-loss of stability in short-crested irregular stern-quartering waves with wind. The capsizing itself was associated with departure from the time-dependent safe basin. The effect of initial conditions was taken into account using statistical correlation between roll and the effective wave. Instead of the Grim effective wave, Vermeer (1990) assumed waves to be a narrow-banded stochastic process. Capsizing is associated with the appearance of negative stability during the time duration, and it is sufficient for a ship to reach a large roll angle under the action of a quasi-static wind load. The probability of capsizing is considered as a ratio of encountered wave cycles, where the wave amplitude is capable of a significant and prolonged deterioration of stability. Themelis and Spyrou (2007) demonstrated the use of the concept of a “critical wave” to evaluate probability of capsizing. Because pure-loss of stability is a one wave event, it is straightforward to separate the dynamical problem from the probabilistic problem. First, the parameters that are relevant to the waves capable of causing pure-loss of stability are searched. Then, the probability of encountering such waves is assessed. To complete the probabilistic problem, the initial conditions are treated in probabilistic sense. Bulian, et al. (2008) combined the effective wave approach with upcrossing theory, where the phenomenon of pure-loss of stability is associated with an upcrossing event for the amplitude of the effective wave. Such an approach allows the time of exposure to be considered directly and produces a time-dependent probability of stability failure. An additional advantage of upcrossing theory is the possibility to characterize the “time above the level”, which in this context is the duration of decreased stability. The mean value of this time can be evaluated by a simple formula, but evaluation of other characteristics is more complex (Kramer and Leadbetter, 1967). In summary, it seems that the second level vulnerability criteria for pure-loss of stability are likely to be probabilistic, due to the very stochastic nature of the

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phenomenon. For criteria related to this mode of stability failure, the time duration while stability is reduced on a wave could be included. Methods useful for addressing pure-loss of stability may include, but are not limited to, the effective wave, the narrow-banded waves assumption, the critical wave approach, and upcrossing theory. As Monte-Carlo simulations may be too cumbersome for vulnerability criteria, at this time, simple analytical and semi-analytical models seem to be preferable.

4 Parametric Roll Parametric roll is the gradual amplification of roll amplitude caused by parametric resonance, due to periodic changes of stability in waves. These stability changes are essentially results of the same pressure and underwater geometry changes that were the cause in the pure-loss of stability mode. However, in contrast to pure-loss of stability, parametric roll is generated by a series of waves of certain frequency. Therefore, parametric roll cannot be considered as a one wave event (Figure 4). a)

20

, deg

0

100

200

300

400

500

100

200

300

400

500

t, s

-20 b)

, deg 200 150 100 50 t, s 0 -50

Fig. 4. Partial (a) and total (b) stability failure caused by parametric resonance

Research on parametric roll began in Germany in the 1930s (Paulling, 2007). In the 1950s, the study of parametric roll for a ship was continued by Paulling & Rosenberg (1959). Paulling, et al. (1972, 1975) observed parametric roll in following waves during model tests in San-Francisco Bay (summary available from Kobylinski & Kastner, 2003). Later it was discovered that this phenomenon could occur in head or near head seas as well (Burcher, 1990; France, et al., 2003).

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Parametrically excited roll motion is at present widely recognized as a dangerous phenomenon, where the cargo and the passengers and the crew may be at risk. Hull forms have evolved from those considered in the development of the 2008 IS Code, as MSC Res. 267(85)— basically the same ship types used 30-50 years ago for the definition of Res. A. 749(18) (2002). For certain ship types, present hull forms appear to have resulted in the occurrence of large amplitude rolling motions associated with parametric excitation. According to this evidence, there is a compelling need to incorporate appropriate stability requirements specific to parametric roll into the new generation intact stability criteria. The simplest model of parametric resonance is the Mathieu equation. It can describe the onset of instability in regular waves. Because the Mathieu equation has a linear stiffness term, it is incapable of predicting the amplitude of roll, and not applicable in irregular waves. Nevertheless, it was used by ABS (ABS, 2004; Shin, et al., 2004) to establish estimates for initial susceptibility to parametric excitation, which corresponds more with intended use for the first level vulnerability criteria. More complex 1(.5)-DOF models include nonlinearity in the time-dependent variable stiffness term. This nonlinearity leads not only to stabilization of roll motion, e.g. establishing finite roll amplitude, but also to a significant alteration of instability boundary in comparison of Ince-Strutt diagram— with unstable motions found outside of the Mathieu instability zone (Spyrou, 2005). This should cause concern about relying on the standard linear stability boundaries. Inclusion of additional degrees of freedom, primarily pitch and heave (Neves & Rodríguez, 2007) results in additional realism. These are especially important for parametric roll in head seas, where pitch and heave may be large. Spyrou (2000) included surging as well. These models, even with their low dimensional character, seem to give reasonable agreement with model tests (Bulian, 2006). Evaluation of the steady-state amplitude of parametric roll can be carried out using asymptotic methods, such as the method of multiple scales (Sanchez & Nayfeh, 1990). Approximate analytical expressions exist for the determination of the nonlinear roll response curve (e.g. ITTC, 2006). The potential of these methods is discussed in Spyrou et al. (2008). The introduction of irregular waves results in drastic changes in the way parametric roll develops. Large oscillations may not be persistent anymore: it may stay, but it may also be transitory (Figure 5). Such behavior has significant influence on the ergodicity of the process— the ability to obtain statistical characteristics from one realization if it is long-enough. Theoretically, the process remains ergodic, as the integral over its autocorrelation function still remains finite, but the time necessary for convergence may be very large. As a result, the term “practical non-ergodicity” is used (Bulian, et al., 2006; Hashimoto, et al., 2006).

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Roll Angle, deg

Roll Angle, deg

40 20 0 -20 -40

Time, s 0

200

400

600

800 1000 1200 1400

0

200

400

600

Time, s 800 1000 1200 1400

40 20 0 -20 -40

Fig. 5. Sample records of simulated parametric roll in long-crested seas (Shin, et al., 2004)

Distribution of parametric roll may be different from Gaussian. Hashimoto, et al. (2006) obtained distributions with large excess of kurtosis from model tests in long- and short-crested irregular seas (Fig. 6). a)

b)

Fig. 6. Distribution of roll angles during parametric resonance in (a) long-crested and (b) short crested seas

Roll damping has more influence on roll amplitude in irregular waves. The damping threshold defines if a wave group can contribute to parametric excitation and therefore, to results in an increase of the roll amplitude. Characteristics of wave groups, such as number, length, and the height of waves in the group, may also have an influence on how fast the roll amplitude increases. In a sense, a regular wave can be considered as an infinite wave group. Therefore, using regular waves could be too conservative for an effective vulnerability criterion, due to the transient nature of this phenomenon (SLF48/4/12). Focusing on the instability boundaries in irregular waves, Francescutto, et al., (2002) used the concept of “effective damping,” also mentioned in Dunwoody (1989a); that is meant to be subtracted from roll damping. Bulian, et al. (2004) used the Fokker-Planck equation and the method of multiple scales for the same purpose. They utilized Dunwoody’s spectral model of GM changes in waves (1989a, 1989b). Furthermore, Bulian (2006) made use of Grim’s effective wave

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concept. Despite overestimating the amplitude of roll, the boundary of instability was well identified. Application of the Markov processes for parametric roll in irregular waves was considered by Roberts (1982). To facilitate Markovian properties, (dependence only on the previous step), white noise must be the only random input of the dynamical system. Therefore, the forming filter, usually a linear oscillator, must be part of the dynamical system to ensure irregular waves have a realistic-like spectrum. These complexities allow the distribution of time before the process reaches a given boundary to be obtained— the first passage method. This method is widely used in the field of general reliability. Application of this approach to parametric roll was considered by Vidic-Perunovic and Jensen, (2009). Time-domain Monte-Carlo simulations (Brunswig, et al., 2006) are increasingly used as computers become more powerful and are integral part of any naval architect’s office. At the same time, caution has to be exercised when using MonteCarlo simulations, due to statistical uncertainty and other methodological challenges that may limit its use for second level vulnerability criteria. Nevertheless, having in mind the difficulties associated with prediction of amplitude for parametric resonance in irregular waves, Monte-Carlo method should not be dismissed. Themelis & Spyrou (2007) used explicit modelling of wave groups as a way to separate problems related to ship dynamics from the probabilistic problem. This method exploited the growth rate of parametric roll due to the encounter of a group, i.e. by solving the problem in the transient stage. Thereafter, the probability of encounter of a wave group can be calculated separately. In principle, this may allow enough simplification to make Monte-Carlo methods into feasible and robust tool for the second level vulnerability criteria for parametric roll. In summary, it seems that the second level vulnerability criteria for parametric roll may be related to the size of the instability area in irregular seas. The group wave approach seems to be promising, as a method to build a criterion based on amplitude.

5 Surf-Riding and Broaching Broaching is the loss of controllability of a ship in astern seas, which occurs despite maximum steering effort. Stability failure caused by broaching may be “partial” or “total”. It can appear as heeling during an uncontrollable, tight turn. This is believed to be intimately connected with the so called surf-riding behaviour. Moreover, it can manifest itself as the gradual, oscillatory build-up of yaw, and sometimes roll, amplitude. The dynamics of broaching is probably the most dynamically complex phenomenon of ship instability. Well known theoretical studies on broaching from the earlier period include Davidson (1948), Rydill (1959), Du Cane & Goodrich (1962), Wahab & Swaan (1964), Eda (1972); Renilson & Driscoll (1981), Motora, et al.(1981); and more recently, e.g. by Umeda &

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Renilson (1992), Ananiev & Loseva (1994), and Spyrou (1996, 1997, 1999). Broaching has also been studied with experimental methods. Tests with scaled radio-controlled physical models have taken place in large square ship model basins; e.g. Nicholson (1974), Fuwa (1982), De Kat & Thomas (1998). Surf-riding is a peculiar type of behaviour where the ship is suddenly captured and then carried forward by a single wave. It often works as predecessor of broaching and its fundamental dynamical aspects have been studied by Grim (1951), Ananiev (1966), Makov (1969), Kan (1990), Umeda (1990), Thomas & Renilson (1990), and Spyrou (1996). Key aspects of the phenomenon are outlined next. In a following or quartering sea, the ship motion pattern may depart from the ordinary periodic response. Figure 7 illustrates diagrammatically the changes in the geometry of steady surge motion, under the gradual increase of the nominal Froude number (Spyrou 1996). An environment of steep and long waves has been assumed. Surf-riding equilibria

Periodic surging

cos(2xG/

At wave trough

At wave crest a

xG Fncr1

b

c Fncr2

d

Fn

Fig. 7. Changing of surging/surf-riding behavior with increasing nominal Froude number

For low Fn, there is only a harmonic periodic response (plane (a)) at the encounter frequency. However, as the speed approaches the wave celerity, the response becomes asymmetric (plane (c)). The ship stays longer in the crest region and passes quickly from the trough. In parallel, an alternative stationary behavior begins at some instance to coexist. This results from the fact that the resistance force which opposes the forward motion of the ship may be balanced by the sum of the thrust produced by the propeller and the wave force along the ship’s longitudinal axis. This is surf-riding and the main feature is that the ship is forced to advance with a constant speed equal to the wave celerity. It has been shown that surf-riding states appear in pairs: one nearer to wave crest and the other nearer to trough. The one nearer to the crest is unstable in the surge direction. Stable surfriding can be realised only in the vicinity of the trough and, in fact, only if sufficient rudder control has been applied.

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Surf-riding is characterised by two speed thresholds: the first is where the balance of forces becomes possible (plane(b)). The second threshold indicates its complete dominance in phase-space, with sudden disappearance of the periodic motion. It has been pointed out that, this disappearance is result of a global bifurcation phenomenon, known in the nonlinear dynamics literature as homoclinic saddle connection (Spyrou, 1996). It occurs when a periodic orbit collides in statespace with an unstable equilibrium— the surf-riding state near to the wave crest. A dangerous transition towards some, possibly distant, undesirable state is the practical consequence. To be able to show the steady motion of a ship overtaken by waves as a closed orbit, one should opt to plot orbits on a cylindrical phase-plane. On the ordinary flat phase-plane the bifurcation would appear as a connection of different saddles (“heteroclinic connection”, see Figures 8 and 9 based on Makov). This is characterised by the tautochronous touching of the “overtaking wave” orbits, running from infinity to minus infinity, with the series of unstable equilibria at crests. This has been confirmed numerically by Umeda (1999). Therefore, no matter what terminology is used, there is consensus and clear understanding, rooted in dynamics, about the nature of phenomena. 20

.

G, m/s

10 Surf-riding

Surf-riding

300

200

Surging

100

0

100

G, m

Surging 10

Fig. 8. Phase plane with surging and surf-riding, Fncr1
The initial attraction into surf-riding, caused by the unstable surf-riding equilibrium near the crest, should be followed by capture of the ship at the stable surf-riding equilibrium near the trough. However, this is a violent transition and, if the rudder control is insufficient, it will end in broaching. The effectiveness of rudder control in relation to the inception of this behaviour, and also characteristic simulations, can be found in the literature (Spyrou, 1996, 1999).

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.

20

G, m/s Surf-riding

10

Surf-riding

G, m 300

200

100

0

100

Surf-riding 10

Fig. 9. Phase plane with surf-riding only Fn>Fncr2

From the above, one sees that, susceptibility to this type of broaching could be assessed by targeting three sequential events: the condition of attraction to surfriding; the condition defining an inability to stay in stable surf-riding, thus creating an escape; and the condition of reaching dangerous heeling during this escape— manifested as tight turn. For the first condition, we already have deterministic (analytical and numerical) criteria based on Melnikov and other methods (Ananiev, 1966; Kan, 1990; Spyrou, 2006). For the second, criteria have been discussed, linking ship manoeuvring indices with control gains (Spyrou, 1996, 1998). The third condition can be relatively easily expressed as the balance of the roll restoring versus the moment of the centrifugal force that produces the turn. A proper probabilistic expression of these conditions is still wanting. However, the critical wave approach may provide a very practical way of dealing with this (Themelis & Spyrou, 2007). Efforts in similar spirit were reported by Umeda (1990) and Umeda, et al. (2007). In reviews of broaching incidents, those that had happened at low to moderate speeds (i.e. far below the wave celerity of long waves) are usually distinguished from the more classical high-speed events. These incidents could have some special importance, because they seem to be relevant also to vessels of larger size, which otherwise are unlikely to broach through surf-riding. Let’s consider now the nature of such a broaching mechanism, which we will call “direct broaching” since the ship is not required to go through surf-riding. It has been shown that, a fold bifurcation is possible to arise for a critical value of the commanded heading, This phenomenon can have a serious influence on the horizontal plane dynamics, creating a stable sub-harmonic yaw (Fig. 10). Further increase of the commanded heading was found to cause a rapid increase of the amplitude of yaw oscillation, leading shortly to a turn backwards of the steady yaw response curve and, inevitably, a sudden and dangerous jump to resonant yaw (Spyrou, 1997). The transient behaviour looks like a growing oscillatory yaw which corresponds closely to what has been described as cumulative-type broaching.

Review of Available Methods for Application to Second Level Vulnerability Criteria

One period motion

One period motion

Fold bifurcation

Rudder angle

Amplitude of periodic motion

One period Stable submotion harmonic yaw

15

Yaw angle

C commanded heading

Flip bifurcation

Stable subharmonic yaw

Fig. 10. Direct broaching

Instability could be an intrinsic feature of the yaw motion of a ship overtaken by very large waves. It has been pointed out that, starting from Nomoto’s equation, the equation of yaw in following waves becomes Mathieu-type (in Spyrou (1997) the derivation and discussion on this and its connection with the above scenario was presented). The amplitude of the parametric forcing term is the ratio of the wave yaw excitation to the hull’s own static gain— maneuvering index K. The damping ratio in this equation receives large values, unless there is no differential control. Due to the large damping, the internal forcing that is required in order to produce instability is much higher (steeper waves) than that at the near zero encounter frequency (i.e. for the mechanism of broaching through surf-riding). Simple criteria connecting the hull with rudder control have accrued from this formulation and could be directly in the vulnerability criteria. A probabilistic formulation has not yet been attempted.

6 Dead Ship Condition Both the term and the concept of dead ship conditions took its root during the steamer era. When a ship looses power, it becomes completely uncontrollable— a significant disadvantage in comparison with ship equipped with sails. The worst possible scenario is a turn into beam seas, where the ship is then subjected to a resonant roll combined with gusty wind. Traditional ship architecture, with a superstructure amidships, made this scenario quite possible. Significant changes in marine technology have led to more reliable engines and a variety of architectural

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types. Nevertheless the dead ship condition— zero speed, beam seas, resonant roll, and wind action, maintains its importance. The present Intact Stability Code (MSC.267(85), 2008), as well as its predecessor (IMO Res. A749(18), 2002), contains the “Severe Wind and Rolling Criterion (Weather Criterion).” This criterion contains a simple mathematical model for ship motion under the action of beam wind and waves. However, some parameters of this model are based on empirical data. Therefore, it may not be completely adequate for unconventional vessels. Thus MSC.1/Circ.1200 (2006) and MSC.1/Circ. 1227 (2007) were implemented, to enable alternative methodologies to be used for the assessment of the Weather Criterion on experimental basis. The problem of capsizing represents the ultimate nonlinearity: the dynamical system transits to motion about another stable equilibrium. When the wave excitation amplitude approaches a critical value, a number of nonlinear phenomena take place. The boundary of the safe basin becomes fractal (Kan & Taguchi, 1991) as a result of the self-crossing branches of an invariant manifold (Falzarano, et al., 1992). Considering the deterioration of the safe basin (Figure 11), Rainey and Thompson (1991) proposed a transient capsize diagram. It uses the dramatic change of the measure of integrity of the safe basin as a boundary value. FINAL CRISIS

FRACTAL BOUNDARY

SMOOTH BOUNDARY 1

d a

Area of SAFE BASIN

b

0

c

e

f

0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized amplitude of excitation

Fig. 11. Deterioration of safe basin with increasing of wave amplitude

Another obvious way to improve criterion is to consider more realistic, e.g. stochastic, models for the environment. This does not necessarily means that the new criterion must be probabilistic. The improvement will be made by adjusting parameters of deterministic criterion to reach similar level of safety for all the vessels which meet it. Studies of probabilistic methods for these criteria were carried out starting in the 1950s (Kato, et al., 1957). As the conventional weather criterion uses an

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energy balance approach, an attempt to consider it from probabilistic point of view seems logical (Dudziak & Buczkowski 1978; a brief description is available from Belenky & Sevastianov, 2007). Similar ideas are behind the Blume criterion (Blume, 1979, 1987), which uses a measure of the remaining area under the GZ curve. Bulian & Francescutto (2006b) and Bulian, et al. (2008) developed a method where the righting arm is locally linearized at the equilibrium, due to the action of a steady beam wind. The probability of large roll is calculated using an “equivalent area” to account for the nonlinearities of the righting arm. The piecewise linear method (Belenky, 1993; Paroka & Umeda, 2006) separates probability of capsizing into two less challenging problems: upcrossings through maximum of the GZ curve and capsizing if such upcrossing has occurred (Figure 12). The probability of upcrossing is well defined in upcrossing theory. While the probability of capsizing after upcrossing can be found using the probability of roll rate at upcrossing that lead to capsizing; this is a linear problem in the simplest case. Statistical linearization can be used to evaluate parameters of the piecewise linear model. Simplicity and a direct relation with time of exposure are among the strengths of the piecewise linear method. 

Upcrossing not leading to capsizing Capsized equilibrium

Threshold

Upcrossing leading to capsizing t

Fig. 12. Separation principle in a piecewise linear method

In principle, a critical wave approach can be used for dead ship conditions as well. However, the outcome of capsizing vs. non-capsizing strongly depends on initial conditions. Therefore, a critical wave group approach similar to Themelis and Spyrou (2007) may be also considered as a candidate for vulnerability criteria.

7 The Choice of Environmental Conditions If vulnerability criterion is deterministic and based on a regular wave assumption, the wave is characterized with height and length. These characteristics must then be related to a corresponding sea state and further on with a safety level. This formulation is not new— this approach is used for evaluation of extreme loads. A typical scheme for calculation of extreme loads is based on long-term assumption, so a number of sea states are considered. An operational profile is

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usually assumed based on existing experience; it includes the fraction of time that a ship is expected to spend in each sea state. Short-term probability of exceedance is calculated for each sea state; then the formula for the total probability is used to determine the life-time probability of exceedance of the given level. This level is typically associated with significant wave height and zero-crossing, or mean period. These data are then directly used to define a regular wave that is expected to cause extreme loads. Similar ideas underlie the reference regular wave in the Weather Criterion in the 2008 IS Code, which was determined from the significant wave steepness, based on the work of Sverdrup and Munk (1947). With all of the similarities, a direct transfer of methods used for extreme loads to stability applications may be difficult. Structural failure is associated with exceeding certain stress levels, actual physical failure is not considered in Naval Architecture. Stresses do not have their own inertia, they follow the load without time lag. The same can be observed about the loads, they occur simultaneously with a wave. They do not depend on initial conditions; even in a nonlinear formulation and do not have multiple responses for the given position of a ship on a wave. Nevertheless, these motion nonlinearities affect only short-term probability, while the scheme as a whole may be applicable. Determining the equivalence between regular and irregular waves represents a problem by itself. The use of significant wave height or steepness has been a conventional assumption, but it does not have a theoretical background behind it. The general problem of equivalency between regular and irregular roll motions was considered by Gerasimov (1979; a brief description in English is available from Belenky & Sevastianov, 2007) in the context of statistical linearization with energy conservation, resulting in an energy statistical linearization. The idea of a regular wave as some sort of equivalent for sea states is attractive because of its simplicity. However, the physics of some stability failures may be sufficiently different in regular and irregular waves. As it was mentioned in the section regarding parametric roll, a regular wave is an infinitely long wave train. Roll damping in this case has a very small influence on the response amplitude, while in irregular seas damping has a significant influence on the variance of response. Therefore, the regular equivalent of irregular waves cannot be applied universally. The application of different environmental models is possible for different stability modes. Surf-riding/broaching is an example of a one-wave event where the concept of a critical wave may fit well. If some of vulnerability criteria are made probabilistic, the next important choice is time scale: long-term or short term. Here, short-term refers to a time interval where an assumption of quasi-stationarity can be applied. Usually this is an interval of three to six hours, where changes of weather can be neglected. Long-term consideration covers a larger interval: a season, a year, or the life-time of a vessel. The short-term description of the environment is simpler and can be characterized with just with one sea state or spectrum. However, if chosen for vulnerability criteria, justification will be required as to why a particular sea must be used. This

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choice is important because sea states which are too severe may make the criterion too conservative and diminish its value. Therefore, special research is needed in order to choose the sea state “equivalent” or “representative” for a ship operational profile. This may naturally result in ship-specific sea-state to use for assessment. An alternative to the selection of a limited set of environmental conditions may be the use of long-term statistics considering all the combinations of weather parameters available from scatter diagrams, or appropriate analytical parametric models (see, e.g. Johannessen, et al., 2002).

8 Summary and Concluding Comments Vulnerability to pure-loss of stability is caused by prolonged exposure to decreased stability on a wave crest, and is likely to be characterized by probabilistic criteria. Methods for a first attempt for vulnerability criteria include the effective wave, a narrow-band or envelope presentation, the critical wave approach, and upcrossing theory. Vulnerability criteria for parametric roll would be more useful if defined in irregular seas. This criterion may be based on size of instability area, while the wave group approach seems to be a good candidate for amplitude-based criterion. Vulnerability to broaching through surf-riding may be judged by the vulnerability to surf-riding. The critical wave approach seems to be the best fit; however, other methods are not excluded. Vulnerability criterion for direct broaching is likely to be deterministic. Vulnerability to stability failure in dead ship conditions may be judged using a variety of methods, as it was the main area of application for both probabilistic and deterministic methods. The modified weather criterion, piecewise linear method, and critical wave or wave group approach may be among the first to be considered. A rational choice of environmental conditions for vulnerability criteria is at least as important as the criteria. An unrealistic environmental condition may lead to incorrect results, even if the criteria are technically correct. However, several possibilities do exist: a regular wave equivalent for life-time risk, a short-term sea state deemed “representative” for a specific ship operational profile, and a longterm approach using a scatter diagram for a representative part of the World Ocean.

9 Acknowledgments The authors are grateful for the funding support from Mr. James Webster (NAVSEA) for this work. The authors are grateful for the guidance and encouragement they received from Mr. William Peters (USCG). The authors also appreciate Dr. Arthur Reed and Mr. Terrence Applebee (NSWCCD) for their help and support.

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References ABS (2004) American bureau of shipping guide for the assessment of parametric roll resonance in the design of container carriers. Houston, Texas Ananiev D M (1966) On surf-riding in following seas. Trans of Krylov Soc 73. Russian Ananiev D M, Loseva L (1994) Vessel’s heeling and stability in the regime of manoeuvring and broaching in following seas. Proc. 5th Intl Conf on Stab of Ships and Ocean Veh. Melbourne, Florida Belenky VL (1993) A capsizing probability computation method. J. Ship Res 37:200-207 Belenky VL, Sevastianov NB (2007) Stability and safety of ships - risk of capsizing (second edition). The Soc of Nav Archit and Mar Eng (SNAME). Jersey City, NJ Belenky V, de Kat JO, Umeda N (2008) Toward performance-based criteria for intact stability. Mar Technol 45:101-123 Blume P (1979) Experimentelle bestimmung von koeffizienten der wirksamen rolldaempfung und ihre anwendung zur abschaetzung extremer rollwinkel. Schiffstechnik, 26 Blume P (1987) Development of new stability criteria for modern dry cargo vessels. Proc. 5th Intl. Symp. on Pract Des of Ships and Other Float Struct. Trondheim, Norway Boroday IK (1967) Statistical characteristics of stability and the probability of capsize of a ship running on any course in irregular seas. Doc of the USSR Expert of the IMCO, Working Gr on Stab of Fish Vessels, IMO. London Boroday IK (1968) Statistical characteristics and probability of capsizing of ship heading with arbitrary course in irregular seas. Trans. of russian register of shipping— theoretical and practical problems of stability and survivability. Transp Publ. Moscow, Russian Boroday IK, Netsvetaev YA, (1969) Ship motion in a seaway. Sudostroenie. Leningrad, Russian Brunswig J, Pereira R, Kim D (2006) Validation of parametric roll motion predictions for a modern ship containership design Proc. 9th Intl. Conf. on the Stab of Ships and Ocean Veh. Rio de Janeiro, Brazil 157-168 Bulian G (2006) Development of analytical nonlinear models for parametric roll and hydrostatic restoring variations in regular and irregular waves. PhD Thesis. Univ of Trieste Bulian G, Francescutto A (2006) On the effect of stochastic variations of restoring moment in long-crested irregular longitudinal sea. Proc 9th Int Conf. on Stab of Ships and Ocean Veh. Rio de Janeiro, Brazil. 1:131-146 Bulian G, Francescutto, A Lugni C (2004) On the nonlinear modeling of parametric rolling in regular and irregular waves. Int Shipbuild Prog. 51 2/3:173-203 Bulian G, Francescutto, A Lugni C (2006) Theoretical, numerical and experimental study on the problem of ergodicity and ‘practical ergodicity’ with an application to parametric roll in longitudinal long crested irregular sea. Ocean Eng, 33:1007-1043 Bulian G, Francescutto A, Maccari A (2008) Possible simplified mathematical models for roll motion in the development of performance-based intact stability criteria– extended and revised version. University of Trieste Burcher RK (1990) Experiments into the capsize of ships in head seas. Proc 4th Int Conf on Stab of Ships and Ocean Veh, Naples, 82-89 Davidson KSM (1948) A note on the steering of ships in following seas. Proc 7th Int Congr of Appl Mech. London, 554-556 DeKat JO, Thomas WL (1998) Extreme rolling, broaching and capsizing— model tests for validation of numerical ship motion predictions. Proc 22nd Symp on Nav Hydrodyn. Washington, DC DuCane P, Goodrich GJ (1962) The following sea, broaching and surging. Trans. RINA 104 Dunwoody AB (1989a) Roll of a ship in Astern Seas—metacentric height spectra. J Ship Res, 33 3:221-228 Dunwoody AB (1989b) Roll of a ship in Astern Seas –response to GM fluctuations. J Ship Res, 33 4:284-290

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Dudziak J, Buczkowski A (1978) probability of ship capsizing under the action of the beam wind and sea as a background of stability criteria. Pol Regist of Ships. Prace StudialnoRoswojowe, Zeszyt 13, Gdansk Eda H (1972) Directional stability and control of ships in waves. J Ship Res 16 3 Falzarano JM, Shaw SW, Troesch AW (1992) Application of global methods for analyzing dynamical systems to ship rolling motion and capsizing Int J of Bifurcation and Chaos, 2 1:101-115 France WG, Levandou M, Treakle TW, Paulling JR, Michel RK, Moore C (2003) An investigation of head seas parametric rolling and its influence on container lashing systems. Mar Technol, 40 1:1-19 Francescutto A (2004) Intact ship stability: the way ahead. Mar Technol, 41:31–37 Francescutto A (2007) Intact stability of ships– recent developments and trends. Proc 10th Int Symp on Pract Des of Ships and Other Float Struct PRADS2007. Houston 1:487-496 Francescutto A., Bulian, Lugni C. (2002) Nonlinear and stochastic aspects of parametric rolling modelling. Proc of 6th Intl. Ship Stab Workshop, Webb Institute, N.Y. also available from Mar Technol, 41 2:74-81 Fuwa T, Sugai K, Yoshino T, Yamamoto T (1982) An experimental study on broaching of a small high speed boat. Papers of Ship Res Inst 66 Gerasimov AV (1979) Energy-statistical theory of nonlinear irregular ship motion. Sudostroenie. Leningrad Russian Grim O (1951) Das schiff in von achtern auflaufender see. Jahrb Schiffbautech Ges 4: 264-287 Grim O (1961) Beitrag zu dem problem der sicherheit des schiffes im seegang schiff und hafen 6:490-497 Helas G (1982) Intact stability of ships in following waves. Proc of Int Conf on the Stab of Ships and Ocean Veh. Tokyo, Japan International Towing Tank Conference - ITTC (2006) Predicting the occurrence and magnitude of parametric rolling— recommended procedure 7.5-02-07-04.3. Rev.01. Effective Date: 2006 Johannessen K, Meling TS, Haver S (2002) Joint distribution for wind and waves in the northern north sea. Int J of Offshore and Polar Eng 12 1 :1-8 Hashimoto H, Umeda N, Matsuda A, Nakamura S (2006) Experimental and numerical studies on parametric roll of a post-panamax container ship in irregular waves. Proc 9th Int Conf on Stab of Ships and Ocean Veh. Rio de Janeiro, Brazil 1:181-190 Kan M (1990) Surging of large amplitude and surf-riding of ships in following seas. Sel Pap Naval Archit Ocean Eng. Soc Nav Archit. Japan 28 Kan M, Taguchi H (1991) Chaos and fractal in capsizing of a ship. Proc of Int Symp on Hydroand Aerodyn in Mar Eng HADMAR’91. Varna, Bulgaria 1:81-88 Kato H, Motora S, Ishikawa K, (1957) On the rolling of a ship in irregular wind and wave. Proc Symp on Beh of Ships in a Seaway, September. Wageningen 1:43-58 Kuo C, Vassalos D, Alexander JG (1986) Incorporating theoretical advances in usable ship stability criteria. RINA Int conf of SAFESHIP project. Ship Stab and Saf. London Kobylinski LK, Kastner S (2003) Stability and safety of ships: regulation and operation. Elsevier, Amsterdam Kramer H, Leadbetter MR (1967) Stationary and related stochastic processes. John Wiley. NY Krylov AN (1958) Selected papers. USSR Acad of Science. Moscow Makov YL (1969) Some results of theoretical analysis of surf-riding in following seas. Trans of Krylov Soc “Maneuverability and Seakeeping of Ships”. Leningrad 126:124-128. Russian Motora S, Fujino M, Koyonagi M, Ishida S, Shimada K, Maki T (1981) A consideration on the mechanism of occurrence of broaching-to phenomena. Trans JSNA 150 MSC.1/Circ.1200, 2006, “Interim guidelines for alternative assessment of the weather criterion. 24 May IMO. London UK MSC.1/Circ.1227, 2007, “Explanatory notes to the interim guidelines for alternative assessment of the weather criterion” 11 January IMO. London UK

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MSC.1/Circ.1228, 2007, “Revised guidance to the master for avoiding dangerous situations in adverse weather and sea conditions” 11 January, IMO. London UK MSC. Res. .267(85), 2008, “adoption of the international code on intact stability, 2008 (2008 IS Code)” 4 Dec IMO. London UK Nechaev YI (1978) Stability of ships in following seas. Sudostr. Leningrad Russian Nechaev YI (1989) Modelling of ship stability in waves. Sudostr. Leningrad Russian Neves MAS, Rodríguez CA (2007) Nonlinear aspects of coupled parametric rolling in head seas. Proc 10th Int Symp on Prac Des of Ships and Other Float Struct. Houston Nicholson K (1974) Some parametric model experiments to investigate broaching-to. Int Symp Dynam Mar Veh and Struct in Waves. London. Palmquist M (1994) On the statistical properties of the metacentric height of ships in following seas. Proc 5th Int Conf on Stab of Ships and Ocean Veh. Melbourne Florida Paroka D, Umeda N (2006) Capsizing probability prediction for a large passenger ship in irregular beam wind and waves: comparison of analytical and numerical methods. J. Ship Res 50:371–377 Paulling JR, Rosenberg RM (1959) On unstable ship motions resulting from nonlinear coupling. J Ship Res 3(1):36-46 Paulling JR (1961) The transverse stability of a ship in a longitudinal seaway j ship research, 4 4 :37-49 Paulling JR, Kastner S, Schaffran S (1972) Experimental studies of capsizing of intact ships in heavy seas. U.S. Coast Guard, Tech Rep (Also IMO Doc. STAB/7, 1973). Paulling JR, Oakley OH, Wood PD (1975) Ship capsizing in heavy seas: the correlation of theory and experiments. Proc of 1st Int Conf on Stab of Ships and Ocean Veh. Glasgow Paulling JR (2007) On parametric rolling of ships. Proc 10th Int Sym on Pract Des of Ships and Other Float Struct. Houston Pollard J, Dudebout A (1892), Theorie du navire. 3. Paris Rainey RCT, Thompson JMT (1991) The transient capsize diagram— a new method of quantifying stability in waves. J Ship Res 35 1 :58-62 Renilson MR, Driscoll A (1982) Broaching – an investigation into the loss of directional control in severe following seas. Trans RINA 124 Res. A. 749(18) as amended by Res. MSC.75(69), 2002, “Code on intact stability for all types of ships covered by imo instruments” IMO. London UK Roberts JB (1982) Effect of parametric excitation on ship rolling motion in random waves. J Ship Res 26:246-253 Rydill L J (1959) A linear theory for the steered motion of ships in waves. Trans RINA 101 Sanchez NE, Nayfeh AH (1990) Nonlinear rolling motions of ships in longitudinal waves. Int Shipbuild Prog 37 411:247-272 Shin YS, Belenky VL, Paulling JR, Weems KM, Lin WM (2004) Criteria for parametric roll of large containerships in longitudinal seas trans. SNAME 112 SLF48/4/12 (2005) On the development of performance-based criteria for ship stability in longitudinal waves. Submitt by Italy, IMO, London SLF50/4/4 (2007) Framework for the development of new generation criteria for intact stability. Submitt by Japan, Netherlands, and USA, IMO, London SLF50/4/12 (2007) Comments on SLF 50/4/4. Submitt by Italy, IMO, London SLF50/INF.2 (2007) Proposal on additional intact stability regulations. Submitt by Germany, IMO, London SLF50/WP.2 (2007) Revision of the intact stability code - report of the working group (part I). IMO, London SLF51/WP.2 (2008) Revision of the intact stability code - report of the working group (part I). IMO, London SLF51/4/1 (2008) Report of the intersessional correspondence group on intact stability. Submitt by Germany, IMO, London

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Spyrou KJ (1996) Dynamic instability in quartering seas: The behaviour of a ship during broaching J. Ship Res 40 1 :46-59 Spyrou KJ (1997) Dynamic instability in quartering seas-Part III: Nonlinear effects on periodic motions. J Ship Res 41 3:210-223 Spyrou KJ (1999) Annals of Marie–Curie fellowships, edited by the European commission. http://www.mariecurie.org/annals/volume1/spyrou.pdf Spyrou KJ (2000) On the parametric rolling of ships in a following sea under simultaneous nonlinear periodic surging. Phil Trans R Soc London, A 358:1813-1834 Spyrou KJ (2005) Design criteria for parametric rolling. Ocean Eng Int, Canada 9 1:11-27 Spyrou KJ (2006) Asymmetric surging of ships in following seas and its repercussions for safety. Nonlinear Dyn, 43:149-172 Spyrou KJ, Tigkas I, Scanferla G, Themelis N (2008) Problems and capabilities in the assessment of parametric rolling. Proc 10th Int Ship Stab Workshop. Daejeon, Korea 47-55 Sverdrup HU, Munk WH (1947) Wind, sea and swell, theory of relations for forecasting. Hydrogr Off Publ 601 Themelis N, Spyrou KJ (2007) Probabilistic assessment of ship stability trans. SNAME, 115:181-206 Thomas G, Renilson M (1992) Surf-riding and loss of control of fishing vessels in severe following seas. Trans RINA 134:21-32 Umeda N (1990) Probabilistic study on surf-riding of a ship in irregular following seas. Proc of 4th Int Conf onn Stab of Ships and Ocean Veh. Naples 336-343 Umeda N, Renilson MR (1992) Broaching— A dynamic analysis of yaw behavior of a vessel in a following sea. In Wilson PA (ed) Maneuvering and Control of Mar Craft, Comput Mech Publ. Southampton 533-543 Umeda N, Yamakoshi Y (1993) Probability of ship capsizing due to pure loss of stability in quartering seas. Nav Archit and Ocean Eng: Sel Pap Soc of Naval Arch of Japan 30:73-85 Umeda N, Shuto M, Maki A (2007) Theoretical prediction of broaching probability for a ship in irregular astern seas. Proc 9th Int Ship Stab Workshop. Hamburg Umeda N, Hori M, Hashimoto H (2007) Theoretical prediction of broaching in the light of local and global bifurcation analysis. Int Shipbuild Prog 54 4:269-281 Vermeer H (1990) Loss of stability of ships in following waves in relation to their design characteristics. Proc of 4th Int Conf on Stab of Ships and Ocean Veh. Naples Vidic-Perunovic J, Jensen JJ (2009) Estimation of parametric rolling of ships – comparison of different probabilistic methods. Proc 2nd Int Conf on Mar Struct. Lisbon, Portugal Wahab R, Swaan WA (1964) Course-keeping and broaching of ships in following seas. J Ship Res 7 4

A Basis for Developing a Rational Alternative to the Weather Criterion: Problems and Capabilities K.J. Spyrou School of Naval Architecture and Marine Engineering, National Technical University of Athens, 9 Iroon Polytechneiou, Zographos, Athens 15780, Greece

Abstract

The feasibility of developing a practical ship dynamic stability criterion based on nonlinear dynamical systems’ theory is explored. The concept of “engineering integrity” and Melnikov’s method are the pillars of the current effort which can provide a rational connection between the critical for capsize wind/wave environment, the damping and the restoring characteristics of a ship. Some discussion about the dynamical basis of the weather criterion is given first before describing the basic theory of the new method. Fundamental studies are then carried out in order to ensure the validity, the potential and the practicality of the proposed approach for stability assessment of ships in a wider scale.

1 Introduction The weather criterion adopted as Resolution A.562 by IMO’s Assembly in 1985 was a leap beyond the “statistical approach” of the earlier Rahola-type general intact ship stability criteria of 1969 where the safety limits were based empirically on GZ characteristics of vessels lost even a 100 years ago (IMO 1993). Despite the criticism the weather criterion has a rational basis in the sense that there is some account of ship roll dynamics integrated within the stability assessment process. Nonetheless this analysis has a simplified character and in the light of recent research advances on the one hand and new trends in design on the other, it seems that the time has come for looking more seriously into the potential of alternative approaches. It is no surprise that at IMO the discussion about the development of new criteria has been re-opened. In the current paper a few steps are taken towards clarifying whether the concept of “engineering integrity” of dynamical systems could be used for setting up an improved and “general purpose” stability criterion (Thompson 1997). This corresponds basically to implementing a M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_2, © Springer Science+Business Media B.V. 2011

25

26 K.J. Spyrou

geometrical approach addressing global behavior and going beyond the ordinary simulation (see for example Falzarano et al 1992). The principal aim is to characterise system robustness to critical environmental excitations. Whilst these ideas have attracted the interest of researchers in the field of ship stability in most cases they have relied on simple generic forms of roll restoring. Hence the question whether such an approach could be used for practical stability assessment remains still unanswered.

2 Basics of the Weather Criterion The IMO version of the weather criterion follows closely Yamagata’s “Standard of stability adopted in Japan” which was enacted as far back as 1957 (Yamagata 1959) while the basic idea had appeared in Watanabe (1938) or perhaps even earlier. The ship is assumed to obtain a stationary angle of heel θ0 due to side wind loading represented by a lever lw1 which is not dependent on the heel angle and is the result of a 26 m/s wind. “Around” this angle the ship is assumed to perform due to side wave action resonant rolling motion as a result of which it reaches momentarily on the weather side a maximum angle θ1 (Fig. 1). As at this position the ship is most vulnerable to excitations from the weather-side it is further assumed that it is acted upon by a gust wind represented by a lever lw2 = 1.5 lw1. This is translated into a 1.5  1.2247 increase of the experienced wind velocity assumed to affect the ship for a short period of time but at least equal with half natural period under the assumption of resonance. The choice of wind corresponds to extreme storm situation. In fact the criterion corresponds to a kind of average between the centre of a typhoon where the wind is very strong but the rolling is assumed that it is not so violent due to the highly irregular nature of the sea; and the ensuing situation which prevails right after the centre of the typhoon has moved away where the wind velocity is lower but the rolling becomes more intense as the waves obtain a more regular form. As pointed out by Rachmanin in his discussion of Vassalos (1986) the criterion is based on a roll amplitude with 2% probability of exceedance which perhaps corresponds to Yamagata’s choice of roll amplitude θ1 as 70% of the resonant roll amplitude in regular waves. The requirement for stability is formulated as follows: should the ship roll freely from the off-equilibrium position with zero angular velocity the limiting angle θ2 to the lee-side should not be exceeded during the ensuing half-cycle. This limiting angle is either the one where significant openings are down-flooded the vanishing angle θν or the angle of 50 degrees which can be assumed as an explicit safety limit whichever of the three is the lowest. Expressed as an energy balance: the work done by the wind excitation as the ship rolls from the wind-side to the leeside should not exceed the potential energy at the limiting angle θ2.

A Basis for Developing a Rational Alternative to the Weather Criterion

Restoring and heeling lever

a

b lw2

lw1 θ2

0 θ1

27

θc

θ

θ0 Fig. 1 The IMO weather criterion

θ2

θ

Fig. 2 The structure of the weather criterion

A number of extra comments pertain to the modeling of system dynamics: although the input of energy from the waves is taken into account for the calculation of the attained angle θ 1, it is not considered during the final half-cycle and the ship is like being released in still-water from the angle θ1 (Fig. 2). The only energy balance performed concerns potential energies in the initial and final positions. An advantage of the criterion is that the damping is somehow present in the calculation of the resonant roll angle θ 1 both in terms of hull dimensions, form

28 K.J. Spyrou

and fittings. Also, the fact that the energy dissipated through damping during the half-roll is not accounted is not so important because it does not reduce seriously the maximum attained angle and in any case the result is conservative. However, the damping function is taken as a pure quadratic which is today an unnecessary simplification. Finally, as in most critical cases the nonlinear part of the restoring curve “participates” in the dynamics, it appears unreasonable this effect to be left unaccounted in the calculation of θ1 (the only nonlinearity considered concerns the damping function).

c GZC 0

GZmax

θ θdownflooding

25 deg Fig. 3 The naval version of the weather criterion

In 1962 Sarchin & Goldberg presented the “naval version” of the weather criterion which whilst more stringent than the above it adheres to the same principle with only few minor differences. This work is the basis for the stability standards applied from most western Navies nowadays as evidenced by documents such as N.E.S 109 (2000) of the British Navy, DTS 079-1 of the U.S. Navy, NAV-04-A013 of the Italian etc. For ocean-going naval vessels the wind speed is assumed to be 90 knots and it is varying with the square of the heel angle (seemingly an improvement over the straight-line shown in Fig. 1; yet perhaps an equally crude approximation of reality, see the discussion of Prohaska in Yamagata 1959). The amplitude of resonant roll due to beam waves is prescribed to 25 degrees which means that the important connection with the roll damping characteristics of the ship that is found in IMO’s and Yamagata’s criterion is lost. Here however is required that the equilibrium angle (point c in Fig.3) does not exceed 20 degrees and also that the GZ at that point is less than 60% of the maximum GZ. The energy requirement of the criterion prescribes that a substantial margin of potential energy should be available at the limiting angle position in excess of the overturning energy. This is expressed through the well-known relationship A2  1.4 A1 .

A Basis for Developing a Rational Alternative to the Weather Criterion

29

3 The Concept of “Engineering Integrity” Several articles have been published about the concept of “loss of engineering integrity” of dynamical systems. The reader who is not acquainted with the basic theory and terminology is referred for example to Thompson (1997). The basic idea is that the safety robustness of an engineering system can be represented by the integrity of its “safe basin” comprised by the set of initial conditions (in our case pairs of roll angle and velocity) which lead to bounded motion. It is known that the basin could suffer some serious reduction of area (“basin erosion”) once some critical level of excitation is exceeded given the system’s inertial damping and restoring characteristics. The reason for this is the complex intersection of “manifolds” i.e. those special surfaces which originate from, or end on, the unstable periodic orbits corresponding to the vanishing angles which become timedependent when there is wave-forcing. Detailed analysis about these phenomena can be found in various texts see for example Guckenheimer & Holmes (1997). The initiation of basin erosion can be used as a rational criterion of system integrity which in our case and for a naval architectural context would be translated as sufficient dynamic stability in a global sense. The methods that exist for predicting the beginning of basin erosion are the following:

a) Use of the so-called Melnikov analysis which provides a measure of the “closeness” of the critical pair of manifolds and hence can produce the condition under which this distance becomes zero. For capsize prediction the method is workable when the damping is relatively low (strictly speaking it is accurate for infinitesimal damping - however several studies have shown that the prediction is satisfactory for the usual range of ship roll damping values). Ideally the method is applied analytically; or it may be combined with a numerical part when the required integrations cannot be performed analytically. b) Direct numerical identification of the critical combination of excitation damping and restoring where the manifolds begin to intersect each other. This method is more accurate but requires the use of specialised software. c) Indirect identification of the critical condition by using repetitive safe basin plots until the initiation of basin erosion is shown. The same comment as in (b) applies here. In the next Section is provided an outline of Melnikov’s method and application for a family of restoring curves parameterised with respect to bias intensity (e.g. produced by the wind excitation). The critical for capsize wave slope as a function of the wind-induced heel and the damping is determined. In the remaining Sections is targeted the concept of engineering integrity itself. More specifically the following two topics are addressed:

30 K.J. Spyrou

 The value of the concept for higher order restoring curves: the practically interesting point here is to find whether it is workable for arbitrary types of restoring. Towards this the family of 5th order restoring functions is analysed. In addition, the concept is applied for the assessment of an existing ship thus giving an idea of how the method would work for “real-world” stability assessment.  Possible use of the concept in a framework of design optimisation: a simple parameterised family of ship-like hull forms has been considered with main objective to determine whether the method can discriminate meaningfully between good and poor designs.

4 Outline of Melnikov’s Method Assume the dynamical system

x τ   f x   ε g x ,τ 

(1)

Where:

x is in our case the vector x, dx / d  of roll displacement (normalised with respect to the vanishing angle) and roll velocity Ha m

ilto nia n

x f (x)

d(t)

Fig. 4 The safe basin and the distance of manifolds

f x  represents the “unperturbed” Hamiltonian part of the dynamical system; and gx ,  is the “perturbation” which includes the damping and the explicitly time-dependent wave-forcing.  is a parameter that represents the smallness of gx ,  . It can be shown that the closeness of manifolds is expressed as (Fig. 4): d τ 0  

ε Μ τ 0   O ε2 f x 

 

(2)

A Basis for Developing a Rational Alternative to the Weather Criterion

31

Where τ 0 is a phase angle in the range 0  τ 0  2 /  . Since the denominator

is of the order of 1 the function M τ 0  (which is called Melnikov function) is to first order a good measure of the distance of manifolds:

M τ 0  







f x τ   gx τ , τ  τ 0 dτ

(3)

g The wedge symbol  means to take the cross product of the vectors f and . The main intention when this method is applied is to identify the critical combinations of design/operational parameters where the Melnikov function admits real zeros which means that the manifolds begin to touch each other (it is essential the crossing of manifolds to be transversal but we do not discuss this here further).

5 Melnikov’s Method for Ship Rolling With a Wind-Induced Bias We have applied the method for the following family of restoring curves which have the bias as free parameter:

Rx   x1  x (1  ax   x  1  a x 2  ax 3

(4)

The parameter a indicates the strength of the bias assumed here to be due to beam wind loading. Letting x  x1 and x  x 2 the roll motion could be described by the following pair:

x1  x2

x2   x1  1  a x12  ax13  F sin Ωτ  τ 0   β x2 

(5)

If the bias is relatively strong the unperturbed part is represented by the vector:

x2   f x    2 3  x1  1  a x1  ax1  Where

(6)

32 K.J. Spyrou

x  x   1   x2 

(7)

While the perturbation is:

0   g x     F sin Ωτ β x 2 

(8)

After substitution into (3) the Melnikov function takes the form of the following integral:

M  0  





-

x2 F sinΩτ  τ 0   βx2 dτ  M E τ 0   M D

(9)

where M E τ 0  is the part due to wave forcing that is explicitly time-dependent; and M D is the part due to the damping. 0.055 F

Ω=1 0.05 0.045

Ω =0.9

0.04 Ω =0.8 0.035 0.8

0.82

0.84

0.86

0.88

0.9

0.92

a

0.94

Fig. 5 Critical F as function of the bias. To be noted the initial “sharp” fall of the critical F as the bias a departs from 1 which corresponds to a symmetric system

For the calculation of (9) we need to have available an explicit expression in terms of time of the roll velocity x 2 corresponding to the unperturbed system. This is determined as follows: the homoclinic orbit goes through the corresponding saddle point at x = 1. With the coordinate change s  x  1 the equation of the unperturbed system becomes:

A Basis for Developing a Rational Alternative to the Weather Criterion

d 2s dτ 2

 as 3  2a  1s 2  a  1s

33

(10)

With some manipulation the above can be written as:

ds   hs d

s  p 2  q 2

(11)

The parameters that appear in (11) are defined as follows:

h

21  a 2  a  3a

22a  1 a ,p and q  2 3a

(12)

The solution of (11) is:

sτ   xτ   1  ∓

p

2

 q2



(13)

p  qcosh 1  a τ

Differentiation of (13) yields:

 



ds dx q 1  a p 2  q 2 sinh 1  aτ   x2   2 d d p  qcosh 1  aτ







(14)

With substitution of (14) into (9) we can calculate the integral (9) with the method of residues. After some algebra, one obtains that the Melnikov function becomes zero when:





 p h  p 2  q 2 p 2  2q 2  3 pq 2 arccosh  q sinh μπ F β   p 6ππ p 2  q 2 sin  μ arccosh  q 

(15)

The sustainable wave slope is:

 Ak crit. 

Fφ ν ν Ω2

(16)

34 K.J. Spyrou

where φ ν is the vanishing angle and ν 

Ix . Ix  δIx

The critical wave slope obtained with application of the above method is shown in Fig. 5 as a function of the frequency ratio for different values of the bias. Notably Melnikov’s analysis produces an identical result with energy balance where the energy influx due to the forcing is equated with the energy dissipated through damping around the remotest orbit of bounded roll of the corresponding “unperturbed” system. This formulation based on energy balance is intuitively quite appealing. It will be discussed in more detail in Appendix I. Melnikov’s method has been applied also for irregular wave excitation (e.g. Hsieh et al. 1994). In Appendix II we outline the main issues involved in the formulation of Melnikov’s method for a stochastic wave environment.

6 Concept Evaluation for Realistic GZ Here our objective is twofold: firstly to clarify whether the engineering integrity concept is meaningful when applied to higher-order and thus more practical GZ curves. Secondly to apply the new method for an existing ship which satisfies the weather criterion (this work was part of a diploma thesis at NTUA (Papagiannopoulos 2001).

a) Higher-order GZ In the first instance we considered the family of fifth-order restoring polynomials

Rx   x  cx 3  c  1 x 5

(17)

Characteristic GZ shapes produced by (17) as c is varied are shown in Fig. 6. A feature of this family is that it covers hardening ( c  0 ) as well as softening ( c  0 ) type levers. It must be noted that the curves shown in Fig. 5 are scaled which means that there is no need the increase of c to lead to a higher value of the real GZ max . The main purpose of the study was to confirm that there is indeed

A Basis for Developing a Rational Alternative to the Weather Criterion

35

sudden reduction of basin area beyond some critical wave forcing for all the members of this family i.e. that the concept is “robust” with respect to the characteristics of the restoring function.

Fig. 6 Scaled “quintic” restoring curves

In Fig. 7 are shown some typical results of this investigation for an initially hardening GZ at four different levels of damping. The points on the curves are identified with measurement of the basin area and scaling against the area of the corresponding unforced system. The curves confirm that for the fifth-order levers the concept of engineering integrity is applicable without any problem i.e. there is indeed a quick fall of the curve beyond some critical forcing. This critical forcing should be approximately equal to the forcing determined from the Melnikov method. An advantage however of the direct basin plot is that it allows to determine fractional integrities also like 90% 80% etc. as function of damping and excitation (90% integrity means that the remaining basin area is 90% of the initial). This is quite important for criteria development since it provides the necessary flexibility for setting the boundary line; for example one could base the maximum sustainable wave slope at 90% integrity rather than at 100% which might be too stringent.

36 K.J. Spyrou

c =1, Ω=1

integrity 1

0.8

0.6

0.4

0.2 0

0.05

0.1

0.15

0.2

0.25

Forcing amplitude F integrity

c =1, Ω=0.88

1

b 0.01 0.05 0.1 0.2

0.8

0.6

0.4

0.2 0

0.05

0.1

0.15

0.2

0.25

Forcing amplitude F

Fig. 7 Integrity diagrams

b) Application for an existing ship A ferry was selected which had operated in Greek waters with LBP  153 m B = 22.8 m T = 6.4 m Cb = 0.548 and KG  10.004 m. The GZ curve was known while the damping as a function of frequency was determined using the wellknown method of Himeno (1981). For the wave-making part we used the panel code Newdrift (1989) which is available at the Ship Design Laboratory of NTUA. In Fig. 8 is shown the sustainable wave slope for a range of frequencies around resonance on the basis of repetitive basin plotting. From this diagram it is possible to determine the range where survivability in a beam sea condition is ensured. This graph could be contrasted against Fig. 11 of Yamagata (1959) which gives the

A Basis for Developing a Rational Alternative to the Weather Criterion

37

wave steepness considered in the weather criterion as function of natural period under the assumption of resonance condition.

Fig. 8 Critical wave environment for an existing ship on the basis of the new concept

7 Use of the Concept of Engineering Integrity for Design Optimisation The final study reported here was based on another NTUA diploma Thesis (Sakkas 2001). It includes application of the method for a family of simplified hull forms whose offsets are given by the following simple equation:

y   f x , z    X x   Z x , z 

(18)

The same family was used in a paper by Min & Kang (1998) of the Hyundai Heavy Industries for optimising the resistance of high-speed craft. The shape of waterlines is described by the function X x  which depends only on the x nondimensionalised position x  where x is the longitudinal position measured l from the middle of the ship and l is the entrance length with 1  x  1 . The function X x  is expressed as the 4th order polynomial

38 K.J. Spyrou

X x  



B 3 1  a 2 x 2  a3 x  a 4 x 4 2



(19)

Fig. 9 Characteristic hull-shape of the considered family

The transverse hull sections depend on both longitudinal position and height. They are expressed by the following simple function

Z x , z   1  z n  x 

(20)

where

n x   s  t  x

(21)

with s, t free parameters. After a few trials we found that for somewhat realistic hull shapes the parameters a 2 , a3 and a 4 should take values between –1 and 1 while the parameters s and t should lie between 0 and 0.4. As s and t increase the hull becomes more slender. A characteristic hull-shape can be seen in Fig. 9.

A Basis for Developing a Rational Alternative to the Weather Criterion

39

0,08 0,07 0,06 0,05 0,04 0,03 0,02 0,01 0 1

3

5

7

9 11 13 Ship No

15

17

19

Fig. 10 Roll damping for each examined hull

Although originally the height z is non-dimensionalised with respect to the draught, in our case this is done with respect to the depth because the hull-shape above the design waterline is very important in stability calculations ( 1  z  0 ). To narrow further on the free parameters of this investigation we have fixed the length, the beam and the depth; respectively to 150m, 27.2m and 13.5m. The displacement was also kept constant at 23710 t which means that only the block coefficient and the draught were variable. In addition we have considered only hulls fore-aft symmetric. KG was linked to the depth while the windage area was set to be 4.1 L x T. 0,035 0,03

(H / λ) critical

0,025 0,02 0,015 0,01 0,005 0 1

3

5

7

9 11 13 Ship No

15

17

19

Fig. 11 Dynamic stability performance: Left bars are for waves only; while the right bars are for wind and waves.

40 K.J. Spyrou

The free parameters were then varied in a systematic way so that a 2  a3  a 4  1 and s  t . In total 20 ships were collected for further stability investigation. For each one of these ships the corresponding GZ curve was determined by using the commercial programme Autoship while the damping was calculated as described in the previous Section. The roll radius of gyration was calculated with a well-established empirical formula. The nondimensional damping value β at roll resonance for each ship is shown in Fig. 10. For all these simplified ships we applied the weather criterion and we verified that it is fulfilled. Thereafter knowing the restoring damping and inertial characteristics for each hull we run the programme Dynamics of Nusse & Yorke (1998) in order to determine the critical wave slope Ak at which basin erosion is initiated. The wind loading was calculated as prescribed in the weather criterion. The performance of each ship as determined from this procedure is shown in the bar chart of Fig. 11. In the same figure is shown the stability performance of the ships when they were subjected to beam wind loading of the same type and intensity as the considered in the IMO weather criterion. The best hull determined is shown in Fig. 12.

Area (m-rad) 1.5 1.0 GZ (m) 0.5 0 0

50 Heel angle (degrees)

100

Fig. 12 Best hull and the corresponding restoring curve

8 Concluding Remarks An investigation on the concept of engineering integrity and whether it can be used for developing a practical ship stability criterion has been presented. In the spirit of the weather criterion we have confined ourselves to a study of system robustness to combined wind and wave excitations coming from abeam and under the assumption that there are no phenomena such as wave breaking on the ship

A Basis for Developing a Rational Alternative to the Weather Criterion

41

side or accumulation of water on the deck. Also, we have left out of the present context any discussion about stability in following seas, a matter which deserves a separate approach due to the different nature of the dynamics involved. Although the following-sea is the most dangerous environment of ship operation, it was not seriously addressed up to now in the IMO Regulations. As a matter of fact, ships are designed with no requirement for checking their stability at the most critical condition that they may encounter although the primary modes of ship capsize in a following sea environment are now well understood. Of course such a matter is not raised for the first time and in the academic world several approaches have been discussed in the past: some are based on the area under the time-varying GZ curve (butterfly diagram see e.g. Vassalos 1986) while more recently there is a trend for taking into account “more fully” system dynamics. For parametric instability already exist general criteria that have been developed in the field of mechanics. However it is doubted whether these are known to ship designers (for an attempt to summarize these criteria see for example Spyrou et al. 2000). Furthermore the clarification of the dynamics of broaching-to has opened up new possibilities for the development of a simple criterion for avoiding the occurrence of this phenomenon too. It is emphasized that the proposed methodology is a platform for developing also formulations targeting the following-sea environment. Furthermore since this approach is a global method and is not based on ordinary simulation it presents also distinctive advantages over the so-called “performance-based methods” a term which in an IMO-style vocabulary is assumed to mean the execution of a limited number of experimental or simulation runs. Our analysis up to this stage shows that although the proposed method can produce meaningful results there are a few practical problems whose solution is uncertain and would require further research. A deeper look into the Melnikovbased approach is presented in a companion paper (Spyrou et al 2002). Some questions that immediately come to mind are: • How to deal with down-flooding angles or more generally limits of stability that are lower than the vanishing angle? The problem arises from the fact that the method targets the phenomena affecting the basin boundary while the amplitude of rolling may be well below that level if there is flooding of non-watertight compartments. The formulation of Melnikov’s method as an energy balance might give the idea for a practical solution.

• How to combine the deterministic and the stochastic analysis? This happened quite meaningfully in the weather criterion and some comparable yet not obvious in terms of formulation approach is required. • How to adjust the level of stringency of the criterion? This could be achieved perhaps by setting the acceptable level of integrity to a fractional value such as 90% rather than the 100% that is currently considered in research studies.

42 K.J. Spyrou

On the other hand, the method seems that it can easily fulfil the prerequisites of a successful stability criterion which in our own opinion are the following:

• It should have an unambiguous direction of stability improvement i.e. should be formulated in such a way that the designer can maximise the stability margin of a ship under consideration (not in isolation of course but in parallel with other performance and safety matters) and not simply check for conformance to limiting values. • The criterion should be representative of stability in a global sense i.e. should not be based on a prescribed and narrowly defined condition where stability should be ensured. • It should always make clear to the designer the connection with the limiting environmental conditions. • The method should take into account the transient nature of the ship capsize process and it should account with sufficient accuracy for system dynamics. • It should be flexible in order to accommodate changes in design trends i.e. should not be overly dependent on existing ship characteristics which means that the degree “empiricism” intrinsic to the method should be minimal.

References Falzarano J Troesch AW and Shaw SW (1992) Application of global methods for analysing dynamical systems to ship rolling and capsizing. Int J of Bifurc and Chaos 2:101-116. Frey M and Simiu E (1993) Noise-induced chaos and phase-space flux Physica D 321-340. Guckenheimer J and Holmes P (1997) Nonlinear oscillations dynamical systems and bifurc of vector fields. Springer-Verlag N.Y. Himeno Y (1981) Prediction of roll damping – state of the art. Report No. 239. Dep of Naval Archit and Marine Eng The Univ of Michigan Ann Arbor MI. Hsieh SR Troesch AW and Shaw SW (1994) A nonlinear probabilistic method for predicting vessel capsizing in random beam seas. Proc of the Royal Soc A446 1-17. IMO (1993) Code on Intact Stability for All Types of Ships Covered by IMO Instruments. Resolut A.749(18) London. Min KS and Kang SH (1998) Systematic study on the hull form design and resistance prediction of displacement-type-super-high-speed ships. J of Marine Sci and Technol 3: 63-75. NES 109 (2000) Stability Standard for Surface Ships. U.K. Ministry of Defence Issue 1 Publication date 1st April 2000. Nusse HE and Yorke JA (1998) Dynamics: Numerical explorations Spinger-Verlag N.Y. Papagiannopoulos S (2001) Investigation of ship capsize in beam seas based on nonlinear dynamics theory. Diploma Thesis, Dep of Naval Archit and Marine Eng, Nat Technical Univ of Athens Sept. Newdrift Vers. 6 (1989) User’s Manual. Ship Design Laboratory, Nat Technical Univ of Athens. Sakkas D (2001) Design optimisation for maximising dynamic stability in beam seas. Diploma Thesis Dep of Naval Archit and Marine Eng, Nat Technical Univ of Athens Sept.

A Basis for Developing a Rational Alternative to the Weather Criterion

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Spyrou KJ (2000) Designing against parametric instability in following seas. Ocean Eng, 27 625653. Spyrou KJ, Cotton B and Gurd B (2002) Analytical expressions of capsize boundary for a ship with roll bias in beam waves. J of Ship Res 46 3 167-174. Thompson JMT (1997) Designing against capsize in beam seas: Recent advances and new insights. Appl Mech Rev. 50 5 307-325. Vassalos D (1986) A critical look into the development of ship stability criteria based on work/energy balance. R.I.N.A. Transactions, 128, 217-234. Watanabe (1938) Some contribution to the theory of rolling I.N.A. Transactions, 80, 408-432. Wiggins S. (1992) Chaotic transport in dynamical systems Springer-Verlag N.Y. Yamagata, M. (1959) Standard of stability adopted in Japan, RINA Transactions, 101, 417-443.

Appendix I: Melnikov’s Method as Energy Balance Consider again the roll equation with bias:

x  β x  x1  x 1  ax   F cos τ

(I1)

The corresponding Hamiltonian system is:

x  x 1  x 1  ax   0

(I2)

with kinetic energy

KE 

1 2 x 2

(I3)

and potential energy

Fig. I1 Energy balance around the homoclinic orbit

44 K.J. Spyrou

PE  x 1  x 1  ax   c 



1 2 1  a  3 a 4 x  x  x 2 3 4

(I4)

It is obvious that if we take as initial point the saddle (x=1 x  0 ) and as final the same point after going over the homoclinic orbit (“saddle loop”) the difference in potential energy will be zero. Zero will be also the difference in kinetic energy (at the saddle point the roll velocity is obviously zero). The energy lost due to the damping “around” the homoclinic orbit is represented by the area A inside the loop times the damping coefficient which is expressed as (Fig. I1):





DE  β xdx  β x 2 dτ

(I5)

The input of energy on the other hand due to the wave excitation is the integral (in terms of the roll angle) of the external roll moment taken around the homoclinic orbit. Mathematically this is expressed as

WE  F sin  τ  τ 0  dx  F sin  τ  τ 0  x dτ





(I6)

The time constant τ 0 allows to vary the phase between forcing and velocity x . The balance of energies requires therefore that i.e. 

DE  WE  β





x 2 d  F cos  τ  τ 0  x dτ





(I7)



which is identical with (9) when set equal to zero i.e. we have obtained a condition identical to the condition M τ 0   0 described by (3). Despite the clarity of the above viewpoint a note of caution is essential: the above energy balance “works” also for other nonlinear global bifurcation phenomena (e.g. a saddle connection) and application should be attempted only when it is already ensured that the underlying phenomenon is an incipient transverse intersection of manifolds which indeed generates basin erosion.

Appendix II: Formulation for Irregular Seas In the “stochastic” version of Melnikov’s method the objective is to determine the time-average of the rate of phase-space flux that leaves the safe basin. The basic theory can be found in Wiggins (1990) and Frey & Simiu (1993). However the

A Basis for Developing a Rational Alternative to the Weather Criterion

45

specific formulation and adaptation for the ship capsize problem is owed to Hsieh et al (1994). The key idea is that the probability of capsize in a certain wave environment is linked to the rate of phase-flux. The formulation is laid out as follows: Let’s consider once more the roll equation, this time however with a stochastic wave forcing F (τ ) at the right-hand-side. It is noted that the equation should be written in relation to the absolute roll angle because the presentation based on angle relatively to the wave in this case is not practical. It can be shown that the flux rate i.e. the rate at which the dynamical system loses safe basin area scaled by the area A of the basin of the unperturbed system is given by the following expression (Hsieh et al 1994):

⎛ M Φ ε ⎡ = ⎢ H s σ p ⎜⎜ D A Α ⎢⎣ ⎝ Hs σ

⎞ ⎛ M ⎟ + M D P⎜ D ⎟ ⎜H σ ⎠ ⎝ s

( )

⎤ ⎞ ⎟ − M D ⎥ + O ε2 ⎟ ⎠ ⎦⎥

(II1)

where H s is the significant wave height σ is the RMS value of the Melnikov function M (τ ) = M E (τ 0 ) − M D for unity H s and M D is the damping part of the Melnikov function which is commonly taken as non explicitly time-dependent. For wave forcing with zero mean the mean value of the wave part of the Melnikov function is zero thus the mean E [M (τ 0 )] is − M D . Also p, P are respectively the standard Gaussian probability density and distribution functions. The assumption can be made, supported however by simulation studies (Hsieh et al 1994 Jiang et al. 2000), that the flux rate reflects reliably the capsize probability. Conceptually this appears as a direct extension for a probabilistic environment of the connection between basin area and capsizability of a ship under regular wave excitation. The “behaviour” of equation (II1) is that the flux rate increases quickly once some critical significant wave height is exceeded gradually approaching an asymptote as H s tends to infinity. Therefore, someone could set a desirable environment of ship operation in terms of a certain wave spectrum, significant wave height etc. and then through proper selection of the damping and restoring characteristics he could try ensure that his design “survives” in this environment. This is translated in keeping the flux rate as determined by (II1) below a certain limit. Hsieh et al (1994) have suggested to use the point where the asymptote intersects the H s axis as the critical one. The various quantities that appear in (II1) are calculated as follows: For the damping term M D of the Melnikov function it is entailed to determine the integral of some power of the time expression of roll velocity at the homoclinic orbit. In some cases it is possible this to be done analytically. For example for the system described by equations (5) it becomes after calculation:

46 K.J. Spyrou





1  q M D  β h  p 2  q 2 p 2  2q 2  3 pq 2 arccosh  3  p

(II2)

where h, p, q are function of the bias parameter a and they are defined in Section 5.  connects with the spectrum of wave elevation as follows:



 

 2  E M E2 τ 0  



0

   d  2 SM





0

S x  S F   d

(II3)

The spectral density function of the wave forcing is linked to the wave spectrum

S F

   F   2 S   

(this is perhaps the main weakness of the method as this

relationship is valid only for a linear process). The final quantity that needs to be calculated is the power spectrum of the scaled roll velocity

S x   

1 2π





-

xe iτ dτ

(II4)

For the homoclinic loop shown in Fig. I1 this is easily calculated numerically although it can be proven that it receives also an exact analytical solution in terms of hypergeometric functions. With the above we can plot the significant wave height versus, for example, the characteristic wave period which gives a straightforward platform for setting a level of acceptability (e.g. survival up to a certain H s ).

Conceptualising Risk Andrzej Jasionowski, Dracos Vassalos SSRC, NAME, University of Strathclyde / Safety At Sea Ltd

Abstract

Ever present jargon and colloquial notion of risk, coupled with obscuring of the simplicity inherent in the concepts of risk by the sophistication of computer software applications of techniques such as event trees, fault trees, Bayesian networks, or others, seem to be major factors inhibiting comprehension and systematic proliferation of the process of safety provision on the bases of risk in routine ship design practice. This paper attempts to stimulate consideration of risk at the fundamental level of mathematical axioms, by proposing a prototype of a comprehensive yet plain model of risk posed by the activity of ship operation. A process of conceptualising of substantive elements of the proposed risk model pertinent to the hazards of collision and flooding is presented. Results of tentative sensitivity studies are presented and discussed.

1 Introduction It is well recognised that the engineering discipline entails constant decision making under conditions of uncertainty, that is decisions are made irrespective of the state of completeness and quality of supporting information, as such information is never exact, it must be inferred from analogous circumstances or derived through modelling, and thus be subject to various degrees of approximation, or indeed, the information often pertains to problems involving natural processes and phenomena that are inherently random. No better example of such decision making process in engineering can be invoked, than that of development of regulations for the provision of safety. The process of rules development is well recognised to rely on both, (a) qualified “weighting” of a multitude of solutions derived on the basis of scientific research, engineering judgement, experiential knowledge, etc, and (b) debates accommodating the ubiquitous political agendas at play and governing societal concerns. The result of such complex and predominantly subjective pressures is often a simplistic compromise normally in the form of deterministic rules which does not reflect the underlying uncertainty explicitly.

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_3, © Springer Science+Business Media B.V. 2011

47

48

A. Jasionowski and D. Vassalos

Although such standards have formed the main frame of safety provision over a century or so, the acknowledgement of existence of these uncertainties, brought about by the occurrences of accidents with undesirable consequences on one hand, and realisation that economic benefit could be compromised by inbuilt but potentially irrelevant margins, on the other hand, instigated a search such as (SAFEDOR 2005) for better methodologies for dealing with the issues of safety. Notably, such search has been enhanced, or possibly even inspired, by the tremendous progress in computational technologies, which by sheer facilitation of fast calculations have allowed for development of numerical tools modelling various physical phenomena of relevance to engineering practice, at levels of principal laws of nature. Such tools present now the possibility to directly test physical behaviour of systems subject to any set of input parameters, and thus also allow derivation of an envelope of the inherent uncertainty. Existence of such methods, however, is not sufficient for progressing with the development of more efficient means of safety provision, without a robust framework for reasoning on the results of any such advanced analyses undertaken for safety verification processes. Such means evolve naturally from the principles of inductive logic; a field of mathematics dealing with reasoning in a state of uncertainty, (Jaynes 1995) (Cox 1961). In particular the branch comprising probability theory forms a suitable framework for such reasoning in the field of engineering. However, while the concept of probability is known to engineers very widely, it does not seem that the many unresolved to this day subtleties associated with either, its fundamental principles or relevant linguistics, are recognised by the profession. This is the situation especially when a special case of application of the probability theory, namely that of risk, is considered. The uptake of risk assessment process, or more recently strife to develop and implement risk-based design paradigms, (SAFEDOR 2005) (Konovessis 2001) (Guarin 2006), are examples of a new philosophy that could alleviate the aforementioned problems of safety provision, if they can become suitable for routine practice. This in turn, can only be achieved if fundamental obstacles such as lack of common understanding of the relevant concepts are overcome. Therefore, this paper sets to examine some lexical as well as conceptual subtleties inherent in risk, offers some suggestions on more intuitive interpretations of this concept, and puts forward a proposal for risk modelling process as a contributory attempt to bringing formalism in this development.

2 The Risk Lexicon As has been the case with the emergence of probability in the last 350 years or so, (Hacking 1975), the terminology on risk has not evolved into anything that can be regarded as intuitively plausible lexicon, as neither have the pertinent syntax nor semantics been universally endorsed.

Conceptualising Risk 49

Any of tautological phrases, such as: “level of safety standard”, “hazard threatening the safety”, semantically imprecise statements such as: “collision risk”, “risk for collision”, “risk from collision”, “risk of collision”, ”risk from hazard aspects of …”, simple misnomers such as “safety expressed as risk”, “safety risk”, or philosophically dubious slogans such as “zero risk1”, can be found in the many articles discussing or referring to the concepts of risk. A few other, easily misconceiveable terms used in this field can be mentioned: (a) (b) (c) (d) (e) (f) (g)

likelihood, chance, probability frequency, rate uncertainty, doubt, randomness risk, hazard, danger, threat analysis, assessment, evaluation risk control measures, risk control options safety goals, safety objectives, safety functional statements, safety performance, relative safety

Although some preamble definitions of these and other terms are offered from article to article, lack of generic coherence in discussions of the risk concepts among the profession is not helpful in promulgating this philosophy as of routinely quality. This article makes no pretence of having resolved all these subtleties. It merely attempts to adopt a scheme of thinking, which appears to allow for systematic understanding of the concepts of safety and risk.

2.1 Safety By way of introduction of this scheme, the term underlying all this discussion, namely safety, shall be presented to formally set the terms of reference. Without going into intricacies of how complex the definition of the term “safety” can be, it is hereby emphasised that for all engineering purposes and currently emerging wider acceptance of risk concepts, the following definition of safety, (Vassalos 1999), should be adopted widely:

1 The Cournot’s lemma stating that “an event of small probability will not happen” has been initially endorsed by many mathematicians, even acclaimed as the physical “logic of probable” by some, however, the concept has been dismissed for the latter part of the last century on the grounds of Bayesian interpretations of probability, i.e. that “zero” is subject to “personal” interpretation. Hence a statement of “zero risk” remains a philosophical conjecture with little, or indeed, no physical justification.

50

A. Jasionowski and D. Vassalos

safety is the state of acceptable risk This definition is unambiguous and it is practical. It emphasizes explicitly that safety is a state. It thus cannot be calculated, it can only be verified by comparison of the actually established quantity of risk with the quantity of risk that is considered acceptable by relevant authoritative bodies; a criterion or standard, in other words. Thus, it follows that statements such as “higher safety” are incompatible with this definition, as the safety is either attained or it is not and no grades of safety can be distinguished. Appropriate statement would be “higher safety standard”, which directly implies lowered levels of risk as acceptable. Also, expressions such as “safety performance” become semantically imprecise unless, contrary to intuitive perception, the word “performance” implies a binary mode of compliance/no compliance with the set standard, rather than a scale, which in turn, relates to risk not safety. This definition also establishes clear distinction between risk and safety. Risk is the benchmark vehicle to demonstrate a state of safety or lack of it. Thus, safety cannot be “expressed as risk”. What the definition does not resolve is the consistency of the relevant syntax which has come to be used in the every day communication on safety. For instance, expressions such as “fire safety” and “ship safety” imply fundamentally different meanings despite use of the same syntax. While the interpretation analogous to that of the colloquial speech examples such as “orange juice” and “baby juice”, could be accepted or taken for granted, as it is done today, such subtleties should be addressed and resolved for avoidance of any ambiguities in engineering applications, especially when many new concepts such as safety goals or others are being introduced. This, however, is beyond the scope of the discussion of this article. Another worthwhile note should be made here, that this definition is also fully compatible with the universally accepted in today’s world mechanism of safety provision through compliance with regulations. Irrespective of the deterministic or more complex nature of the regulation, safety is said to have been attained, when an acceptable criterion has been met. Although the inherent risk is not disclosed, it is said to have been brought to an acceptable level. Very often expressions, such as “relative safety” or “conditional safety” are used to describe the limited scope of such regulations and undisclosed nature of the risk, though neither of the words “relative” nor “conditional” has any quantifiable explanation. But, as mentioned earlier, since simplistic and disparate regulations could compromise commercial gain or undermine societal approval of the standards proposed for safety provision, when critical accidents happen, new methods are sought after. The direct application of the concept of risk is one such method, especially since the above definition of safety implies the “presence of risks”. The key question now is what is risk?

Conceptualising Risk 51

2.2 Risk An ISO 8402:1995 / BS 4778 standard offers the following definition of risk: “Risk is a combination of the probability, or frequency, of occurrence of a defined hazard and the magnitude of the consequences of the occurrence” A similar definition is put forward at IMO, MSC Circ 1023/MEPC Circ 392: “Risk is a combination of the frequency and the severity of the consequence”. Although rather widely endorsed, these definitions are so ambiguous that they hardly qualify as useful for any engineering purposes. The ambiguity derives from the following: (a) the word “combination” is open to any interpretation whatsoever, both in terms of relevant mathematics as well as underlying physical interpretations (b) the words “probability” and “frequency” have, in common engineering practice, different physical interpretations, which itself can be arbitrary, (c) the words “magnitude” or “severity” are unspecific, (d) the “consequence” is again open to any interpretation. For instance, the syntax of the above definition implies differentiation of the semantics of the phrases such as “hazard” and “consequence”, without any qualification how such differentiation is attained. Thus, the sinking of the vessel can be considered as a hazard, which can lead to the loss of human life, but it can also be considered as an ultimate consequence, or in other words as the loss. See here Fig. 1 and the subsequent discussion. While the merit of proposal of such definitions of risk could be its generic nature, overly lack of any specificity deprives it of practical significance. The following alternative definition is suggested, which is equally generic yet it is comprehendible more intuitively and, apparently, more pragmatic:

risk is a chance of a loss The word “chance” is one of the most fundamental phrases referring to, or being synonymous with the natural randomness inherent in any of human activities and which, in this case, can bring about events that are specifically undesirable, the “loss”. The engineering aspects of the concept of chance are effectively addressed by the fields of probability and statistics, both of which offer a range of instruments to quantify it, such as e.g. expectation, and hence there is no need for introduction of unspecific terms such as “combination”. These well known mathematical tools will be discussed in section 3. This definition of risk resolves also, to some extent, the manner of its classification. Namely reference to risk should be made in terms of the loss rather than in terms of any of the intermediate hazards in the chain of events leading to the loss, a nuance just mentioned.

52

A. Jasionowski and D. Vassalos

Fig. 1 The concept of a chain of events in a sample scenario leading to the loss. Each of the events is a hazard after it materialises. A scenario can be identified by a principal hazard, see Table 1.

For instance, semantics of a statement “risk to life” or “risk of life loss” seem to indicate sufficiently precisely the chance that an event of “fatality” can take place, without much space for any other interpretations. On the other hand, a statement such as “risk of collision” can be interpreted as a chance of an event of two ships colliding; however, given a further set of events that are likely to follow, such as flooding propagation, heeling, capsizing, evacuation/abandoning and fatalities, see Fig. 2, it becomes unclear what is the “loss”, and thus what is the risk? Furthermore, the above definition supports lexically the recently more pronounced efforts to introduce the concept of holism in risk assessment. Namely, rather than imposing a series of criteria for acceptable risk levels in a reductionism manner, whereby separate criteria pertaining to different hazards are introduced, such as e.g. criteria on fire hazards or criteria on flooding hazards (e.g. probabilistic concept of subdivision, (SLF 47/17 2004) ), a unique criterion should be proposed that standardises the acceptable risk level of the ultimate loss, e.g. criteria for loss of life, environmental pollution, etc, and which risks account for each hazard that can result in the loss, a view already mentioned in (Skjong et al. 2005) or (Sames 2005). This concept underlines the risk model proposed in the following section 3.

3 Risk Modelling As mentioned above, the fields of probability and statistics provide all the essential instruments to quantify the chance of a loss.

3.1 Frequency Prototyping To demonstrate the process of application of these instruments for risk modelling, perhaps it will be useful to stress the not necessarily trivial difference between the concepts of probability and frequency. Namely, to assess a “probability of an event”, a relevant random experiment, its sample space and event in question on that sample space must all be well defined. The relative frequency of occurrence of this event, whereby relative implies relative to the sample space, is a measure of probability of that event, since such relative frequency will comply with the axioms of probability.

Conceptualising Risk 53

In the case of “an experiment” of operating a ship, however, whereby events involving fatalities can occur in the continuum of time, the pertinent sample space is customarily not adopted, and rather, the rate of occurrence of these events per unit time is used. The rate can be greater than unity, and obviously can no longer be referred to as probability. Thus, also use of the term “likelihood” would seem to be displaced here since it is synonymous with probability rather than frequency. Therefore, application of the term frequency per ship per year is customary, also since its quantification relies on the historical data for the relevant fleet. Lack of a well defined sample space, however, does not prevent one from using concepts of probability, such as e.g. distribution, expectation, etc, in analyses and analytical modelling. Thus, following from this preamble, it is proposed here that for the purpose of risk modelling the following assumptions are made: (a) The loss is assumed to be measured in terms of an integer number of potential fatalities among passengers that can occur as a result of activity relating to ship operation (b) The loss is a result of occurrence of a set of scenarios which can lead to this loss (c) Scenarios are intersections of a set of events (all must occur) and are identified by principal hazards2, e.g. fire (d) The scenarios are disjoint (if one scenario occurs then the other does not) These assumptions are shown graphically in Fig 2 below.

Fig. 2 Illustration of the concept of a compound event referred to as “fatality” by Venn’s diagram, (Jasionowski 2005), which is a union of mutually exclusive scenarios, whereby each scenario is an intersection of a set of relevant events and is identified by principal hazards, such as collision, fire, etc.

2

an event of a loss scenario is a hazard that materialised

A. Jasionowski and D. Vassalos

54

Deriving from the above, it is hereby proposed that the frequency frN (N) of occurrence of exactly N fatalities per ship per year is derived based on Bayes theorem on total probability, as follows:

frN N    frhz hz j   prN N hz j  n hz

(1)

j 1

Where

nhz is the number of loss scenarios considered, and hzj represents an

event of the occurrence of a chain of events, (a loss scenario), identifiable by any of the following principal hazards: Table 1. Principle hazards

1

Principal hazards, hzj

Average historical frequency of its occurance, frhz(hzj)

Collision and flooding

2.48e-3, (Vanem and Skjong 2004)

2

Fire

-

3

Intact Stability Loss

-

4

Systems failure

-

…, etc.

-

Furthermore, frhz(hzj) is the frequency of occurrence of a scenario hzj per ship per year, and prN(N|hzj) is the probability of occurrence of exactly N fatalities, given loss scenario hzj occurred. In principle, there is nothing new to this proposal, except, perhaps, for the emphasis of the need to estimate the probability for occurrence of exactly N fatalities, prN(N|hzj), conditional on the occurrence of any of the principal scenarios j, an essential element, conceived during (Jasionowski et al. 2005), and often unaccounted for in risk concept proposals, such as e.g. (Sames 2005). Modelling of either of the elements of the equation (1) requires an in-depth analysis of historical data as well as thorough analytical considerations of all reallife processes affecting them, practical attainment of which remains a challenge, requiring inter-disciplinary approach. For a demonstration purposes, a prototype model of the probability of an event of occurrence of exactly N fatalities conditional on the occurrence of a scenario of collision ∩ flooding, that is scenario corresponding to j=1 in Table 1, will be discussed. Namely, it can be shown that prN(N|hzj=1) can be expressed as (2), (Jasionowski et al. 2005). 3 n flood

prN N hz1     wi  p j   ek  ci , j , k  N  i

j

n Hs k

(2)

Conceptualising Risk 55

Where the terms wi and pj are the probability mass functions of three, i=1 ..3, possible ship loading conditions and a j=1 … nflood flooding extents, respectively, assigned according to the harmonised probabilistic rules for ship subdivision, (SLF 47/17 2004), or (Pawlowski 2004). The term ek is the probability mass function 1 derived from (15) for the sea state Hsk, where 0  Hsk  4m , Hsk  k  4  n Hs and nHs is the number of sea states considered. The term ci,j,k(N) is the probability mass function of the event of capsizing in a time within which exactly N number of passengers fail to evacuate, conditional on events i,j and k occurring, and can be tentatively assigned with model (3). t fail  N   t fail  N   ci , j , k ( N )    ln  i , j , k    i , j , k  30   30  

(3)

The term i,,j,k (with y) represents the phenomenon of the capsize band shown in Fig. 3 that is the spread of sea states where the vessel might capsize. These can be estimated as follows:

 Hsk  Hscrit ( sij )         Hs s crit ij  y 

 i , j , k  1  

(4)

 y Hscrit   0.039  Hscrit  0.049

(5)

Where  is the cumulative standard normal distribution. The Hscrit(s) is calculated from equation (16) of Appendix 2. The sij is the probability of survival, calculated according to (SLF 47/17 2004). Its interpretation is discussed in Appendix 2. The concept of a capsize band

6

1 0.9

5

0.7

pdf(Hs)

4

0.6

3

0.5 0.4

2

0.3 0.2

1 0

pcap = cdf(Hs)

0.8

0.1 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0

Hs crititcal

Fig. 3 The concept of “capsize band”, (Jasionowski et al. 2004), for critical sea states of 0.5 and 4.0m.

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t fail  N   t fail  N   t fail  N  1

(6)

t fail  N   N fail1 t 

(7)

N fail t   N max  N evac t 

(8)

Finally, the term Nevac(t) is the number of passengers evacuated within time t , and is referred to as an “evacuation completion curve”, see Fig. 4. Such a curve can effectively be estimated on the basis of numerical simulations, (Vassalos et al. 2001).

Fig. 4 Evacuation completion curve.

It is worthwhile to now offer discussion of some preliminary results from a sample case studies demonstrating risk sensitivity to some key input parameters. The study is discussed below, after introducing the final elements of the risk model in the next section section 3.2.

3.2 Risk as a Summary Statistic The frequency of occurrence of exactly N fatalities per ship year, that is the information provided by the model (1), once all its elements are calculable, allows for direct application of fundamental concepts of statistics for describing the statistical properties of the random events in question, in this case fatalities. The most obvious one is use of a graphical plot of a relationship of this frequency with the number of fatalities. For example, it is very common to graphically plot the cumulative frequency of N or more fatalities, so referred to FN plot, given by equation (9).

Conceptualising Risk 57

FN  N  

N max

 fr i 

iN

(9)

N

Where Nmax is the total number of persons considered (e.g. number of crew, or number of passengers, or both). An example of such relationship derived on the basis of historical data, rather than analytical/numerical modelling, is shown in Figure 5 below.

Fig. 5 Example of an FN plot derived from historical data, (Lawson 2004).

However, while conceptually useful and, indeed, accepted widely as an expression of risk especially when plotted together with related criteria lines, (Skjong et al. 2005), it has been well known that for the purposes of any consistent decision making, (Evans 2003), some form of aggregate information, derived on the basis of such distributions, is required. Commonly used summary statistics, such as expected value, are examples of such aggregate information. It is hereby emphasised that this form of information be used to quantify the “chance” of a loss, or the risk. Unsurprisingly, the expected number of fatalities, E(N), given by (10) and often referred to as the potential loss of life, PLL, has been used among the pertinent profession routinely. See Appendix 1 on the form of equation (10).

Risk PLL  E N  

N max

 F i  i 1

N

(10)

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A. Jasionowski and D. Vassalos

As it is known in the field of statistics, it is always desirable to find a statistic which is not only consistent and efficient, but ideally is sufficient, as then all relevant information is contained in one such number. In case of a statistic which is not sufficient, additional information is required to convey the characteristics of the underlying random process. Since the expectation is hardly ever a sufficient statistic, it is necessary to examine what additional information is required additionally to (10) to quantify risk sufficiently comprehensively. Namely, it is well known that the public tolerability for large accidents is disproportionally lesser than the tolerance for small accidents even if they happen at greater numbers. This tendency is not disclosed in the model (10), as simply the same expected number of fatalities can result from many small accidents or a few larger ones. It is for this reason, that model (10) is also referred to as a risk-neutral or accident-size-neutral, (Evans 2003). Thus, public aversion to large accidents cannot be built into any relevant criteria based on PLL in its current form and without recurs to some form of utility function. Although there are strong suggestions negating such a need and supporting the view that a risk-neutral approach is sufficient, e.g. (Skjong et al. 2005), it does not seem that the above simple fact of real life can be ignored. Indeed, recently proposed criteria for stability based on the so called “probabilistic concept of subdivision” defy this risk-neutral philosophy, as ships with more passengers are required higher index of damage stability. Therefore, a proposal by (Vrijling and Van Gelder 1997) shown as equation (11) seems worth serious consideration as an alternative to (10), as it allows for controlled accommodation of the aversion towards larger accidents through stand. dev.  and a risk-aversion index k.

Risk  E  N   k    N 

(11)

Which form of a statistic is to be used to quantify risk should be the important next step in research in this field. To summarise this chapter, as one can see, risk can be shown to be a statistic of the loss, which can be systematically modelled and used for better informed decision making during any design process. To demonstrate potential benefits of some elements of the concept of a holistic approach to risk modelling and contained in model (1) and (10), simple case studies are discussed next.

4 Sensitivity Studies As has been mentioned earlier, there are no readily available elements related to even a handful of loss scenarios listed in Table 1 and necessary to complete model (1).

Conceptualising Risk 59

However, since it is known that collision and flooding is one of the major risk contributors, a sample study aiming to demonstrate the usefulness of the proposed risk model is undertaken here based on only this loss scenario. Consider a RoRo ship, carrying some 2200 passengers and crew, and complying with the new harmonised rules on damaged ship stability by meeting the required index of subdivision of 0.8 exactly, see Figure 6. Consider also that the evacuation arrangements allow for two different evacuation completion curves, as shown in Figure 7. The question is what is the effect of given evacuation curve on risk to life posed by this ship? The risk is calculated for nhz=1 in (1), using constant frhz(hz1)=2.48•10–3 per ship year. Element (2) is estimated based on the information shown in Figure 6 and Figure 7. Fig. 8 demonstrates the effect of the evacuation completion curve on the conditional probability prN(N|hz1). It seems that the majority of this scenario variants are either no fatalities at all or a very large number of fatalities. Note that because of this trend, the function estimates at these limits determine the ultimate risk, and that the impression of higher overall frequencies for 60-min evacuation curve seen in Fig. 8, must be viewed with care. 1 0.9 0.8 0.7

si

0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.002 0.004 0.006 0.008

0.01

0.012 0.014 0.016 0.018 0.02

pi

Number of passengers evacuated [-]

Fig. 6 Elements of index A, sample case.

2000 60 minutes

120 minutes

1500

1000

500

0

0

20

40

60 80 100 time to evacuaute [minutes]

Fig. 7 Hypothetical evac. completion curve

120

140

A. Jasionowski and D. Vassalos conditional probability mass function of exactly N fatalities prN (N/hz1)

60

1.00E-01 0

500

1000

1500

2000

1.00E-02 1.00E-03

1.00E-04

1.00E-05

1.00E-06 Number of fatalities evacuation complete 60min evacuation complete 120min

Fig. 8 Prob. prN(N|hz1) for sample case

The resultant FN curve derived on the basis of (9) can be seen in the Figure 9. It seems that the distribution is not affected significantly by different evacuation completion curves, though when comparing the risk of 0.881 and 0.939 fatalities per ship year, estimated according to equation (10) for 60 minutes and 120 minutes evacuation completion curves, respectively, a difference of some 6% can be seen as a considerable design/operation target worth achieving.

Fig. 9 FN plot, effect of evacuation completion characteristics on risk

Another interesting test was undertaken, whereby a hypothetical ship with only two flooding extents possible was used together with the 60minutes evacuation completion curve shown in Figure 7. Two cases were assumed for the characteristics of flooding extents, such that in both cases the resultant index A remained the same. Namely in case 1, p11  0.1, s11  0.3000, p12  0.9, s12  0.8560, and for case 2, p12  2.1, s121  0.6000, p22  0.8, s22  0.8505, were assumed. The resultant FN curves curves are shown in Figure 10 below.

Conceptualising Risk 61

Fig. 10 FN plot, effect of subdivision on risk, 60-min evacuation completion curve

As can be seen, contrary to the commonly expressed view, two ships with different subdivision but the same index A, seem unlikely to have the same level of risk.

5 Further Work The process of conceptualising risk discussed in this article is one of the many needed before the vision of Risk Based Design, a design paradigm utilising the concept of risk for the purpose of safety verification, can become a reality. The fundamental elements that need to be in place are (a) an explicit risk model capable of accommodating a number of loss scenarios, sufficient to meaningfully quantify risk, and (b) the criteria, which will reflect both, the societal risk tolerability, and as importantly, the epistemic and the aleatory uncertainties of the proposed risk model.

6 Conclusions This paper discussed the concepts of safety and risk. Many lexical issues, often ignored in the communication on these concepts, have been pointed out. Some suggestions on more intuitive definitions have been made. A process of risk modelling has been explained to some extent, and a simple yet comprehensive model has been presented. A tentative model for probability of exactly N fatalities and conditional on the occurrence of a collision ∩ flooding loss scenario has been presented. Results of calculations of contribution to risk from this scenario have been presented. Preliminary results seem to indicate that the common notion that two different ships with the same index of subdivision correspond to the same level of risk is not justified.

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Acknowledgements The present research was funded by the EC 6th Framework Programme (20022006), project EC FP6 IP 516278 SAFEDOR SP2.1 www.safedor.org. The support received is hereby gratefully acknowledged.

References Comstock JP, Robertson JB (1961) Survival of collision damage versus the 1960 Convention on Safety of Life at Sea. SNAME Transactions, Vol 69, pp.461-522 Cox TR (1961) The algebra of Probable Inference. Oxford Univ Press Evans A W (2003) Transport fatal accidents and F-N curves: 1967-2001. Univ College London, Centre for Transport Studies, Report 073 for HSE Guarin L (2006) Risk Based Design Concept. SAFEDOR, D5.1.1 Hacking I (1975) The emergence of probability. Cambridge Univ Press Jasionowski A (2005) Damage Stability and Survivability. Training Course on Risk-Based Ship Design, TN SAFER EURORO II, Design For Safety: An Integrated Approach to Safe European Ro-Ro Ferry Design, Univ of Glasgow and Strathclyde Jasionowski A, Vassalos D, Guarin L (2004) Theoretical Developments On Survival Time PostDamage. Proc 7th Int Ship Stab Works, Shanghai Jasionowski A, Bulian G, Vassalos D et al (2005) Modelling survivability. SAFEDOR, D2.1.3 Jaynes E T (1995) Probability Theory: The Logic of Sci Washington Univ Konovessis D (2001) A Risk-Based Design Framework For Damage Survivability Of Passenger Ro-Ro Ships. PhD thesis, Univ of Strathclyde, Glasgow Lawson D (2004) Engineering disasters – Lessons to be learned. John Wiley, ISBN 1860584594 MSC 72/16 (2000) Decision parameters including risk acceptance criteria Pawlowski M (2004) Subdivision and damage stability of ships. Euro-MTEC series, Technical Univ of Gdansk SAFEDOR (2005) Risk-Based Design, Operation and Regulation for Ships. EC FP6 IP 516278, www.safedor.org Sames P (2005) Risk Based Design Framework: Design Decision Making. SAFEDOR, D5.1.2, Rev 7 Skjong R, Vanem E, Oyvind E (2005) Risk Evaluation Criteria. SAFEDOR, D4.5.2, DNV SLF 47/17 (2004) Sub-Committee On Stability And load Lines And On Fishing Vessels Safety Tagg R, Tuzcu C (2002) A Performance-based Assessment of the Survival of Damaged Ships – Final Outcome of the EU Research Project HARDER. Proc 6th Int Ship Stab Workshop, Webb Inst Vanem E, Skjong R (2004) Collision and Grounding of Passenger Ships – Risk Assessment and Emergency Evacuations. Proc 3rd Int Conf on Collision and Grounding of Ships, Izu Vassalos D (1999) Shaping ship safety: the face of the future. J. Marine technol36 (2):61-74 Vassalos D, Kim H, Christiansen G, Majumder J (2001) A Mesoscopic Model for Passenger Evacuation in a Virtual Ship-Sea Environment and Performance Based Evaluation. Pedestrian and Evacuation Dynamics, Duisburg Vassalos D, Jasionowski A, Guarin L (2005) Passenger Ship Safety – Science Paving the Way. Proc 8th Int Ship Stab Workshop, Istanbul Vrijling JK, Van Gelder PHAJM (1997) Societal Risk And The Concept Of Risk Aversion. Dep of Civil Eng, Delft Univ Of Technol, Delft

Conceptualising Risk 63

Appendix 1 A Note on the Expected Rate of Fatalities It can be shown that the area under the FN curve represents the expected number of fatalities per ship year, (Vrijling and Van Gelder 1997): N max

N max N max

i 1

i 1 j  i

 FN i  





fN  j 

N max i

 f i  i 1 j 1

N

N max

(12)

 i  f i   E N   PLL i 1

N

Appendix 2 A Note on the Interpretation of the Probability of Survival The survival factor “si” in (SLF 47/17), represents the probability of an event that a vessel survives after flooding of a set of compartments i due to a collision with another vessel and subsequent flooding. The question is what does “survive” mean? The following is the proposed interpretation. The relationship between the survival factors, considering only the final stage of flooding, and the ship parameters are: 1

si , final

Range  4  GZ  K   max  16   0.12

(13)

This has been derived from (14), i.e. the relationship between the parameters GZmax and Range and the critical sea state3, Hscrit, established through physical model experiments, (Tagg and Tuzcu 2002); and the cumulative distribution function F Hscollision of the sea states recorded at the instant of collision and given by equation (15), see also Fig. 11. Note, that for the purpose of derivation of (13), it was implied that Hscrit ~ Hscollision, whereby the relationship between (14) and (15) was obtained through regression analyses rendering directly formula (13).

 GZ max Range  Hscrit  4     16   0.12

(14)

3 a sea state causing the vessel capsizing during about half of the 30minutes scaled model tests, the damage opening modelled was that known as SOLAS damage

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A. Jasionowski and D. Vassalos

FHscollision  e  e

0.16 1.2 Hscollision

(15)

It is worth now offering the form of hard numbers in explaining the meaning of the probability (13), given the underlying relationships (14) and (15). One such form could read as follows: should the vessel suffer 100 collision incidents, say in her lifetime, always leading to flooding of the same spaces “ i ” with a survival factor of, say, si,final = 0.9, then 90 of these incidents will take place in a sea state with significant wave height below Hs(si, final)=2m, see equation (16), with the vessel remaining afloat for at least 30minutes at a relevant damage-state equilibrium (in other words the vessel will “survive”). The remaining 10 of these incidents will take place in a sea state with the significant wave height above Hs(si, final)=2m, and the vessel will not be able to sustain this for the 30 minutes after collision (the vessel will not “survive”), see the note on critical sea state3.

Fig. 11 CDF of significant wave heights at instants of collisions, eq.(15), (Tagg and Tuzcu 2002).

This interpretation follows the “philosophy” behind the formulation for the si,final, discussed in (Tagg and Tuzcu 2002), currently underlying the new harmonised Chapter II Part B of the SOLAS Convention, (SLF 47/17 2004). The relationship between the significant wave heights at the instant of collision, here equal to the critical sea state3, and the “ s ” factors is given by equation (16), (Tagg and Tuzcu 2002), which is derived from equation (15).

Hscrit ( s)  Hscollision ( s ) 

0.16  ln  ln s  1.2

(16)

The aforementioned time of 30 minutes derives from the re-scaled model test duration assumed during the campaign of physical model experiments of the HARDER project, (Tagg and Tuzcu 2002), which underlines the relationship between ship parameters and the sea states3 leading to capsize.

Evaluation of the Weather Criterion by Experiments and its Effect to the Design of a RoPax Ferry Shigesuke Ishida, Harukuni Taguchi, Hiroshi Sawada National Maritime Research Institute

Abstract

The guidelines of experiments for alternative assessment of the weather criterion in the intact stability code were established in IMO/SLF48, 2005. Following the guidelines, wind tunnel tests and drifting tests for evaluating wind heeling lever, lw1, and roll tests in waves for evaluating the roll angle, 1 , were conducted. The results showed some difference from the current estimation, for example the wind heeling moment depended on heel angle and the centre of drift force existed higher than half draft. The weather criterion was assessed for allowable combinations of these results and the effect of experiment-supported assessment on the critical KG and so forth was discussed.

1 Introduction In 2005, the IMO Sub-Committee on Stability, Load Lines and Fishing Vessels Safety (SLF) restructured the Intact Stability Code (IS Code, IMO, 2002), and the weather criterion (Severe wind and rolling criterion), defined in section 3.2 of the code, was included in the Mandatory Criteria (Part A) of the revised code (IMO, 2006). The necessity of the criterion has been recognized to ensure ship stability safety in “dead ship condition”, in which the ability to control the ship is lost. However, the applicability of the criterion to some types of ships (e.g. modern large passenger ships), which did not exist at the time of development of the criterion, have been questioned. In order to solve the problem, the alternative assessment with model experiments is mentioned in the revised code. To ensure uniform applicability of model experiments, which evaluate the wind heeling lever and the resonant roll angle, the guidelines were developed and included as Annex 1 in the revised code. However, they were set as “interim guidelines” because the feasibility, reliability and so forth are not fully clarified and it is

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_4, © Springer Science+Business Media B.V. 2011

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66 S. Ishida et al.

recognized that a considerable accumulation of the experimental experience is required to correctly evaluate the safety. Some effects of this assessment were already discussed (Bulian et al., 2004, Francescutto et al., 2004). However, they were not based on full experiments included in the guidelines. With this background the authors conducted experiments with a Ro-Pax ferry model following the guidelines and examined the above mentioned items. The previous paper (Taguchi et al, 2005, hereafter just referred as “previous paper”) reported the results except the wind tunnel tests. In this paper, the effects of this experiment-supported assessment by full experiments are reported. In the following chapters the items explained in the previous paper are mentioned concisely.

2 The Weather Criterion and its Alternative Assessment

Lever

The weather criterion evaluates the ability of a ship to withstand the combined effects of beam wind and waves. The criterion requires that area “b” should be equal to or greater than area “a” (see Fig. 1), where lw1 : steady wind heeling lever at wind speed of 26 m/s lw2 : gust wind heeling lever (lw2 = 1.5 lw1) 1 : roll amplitude in beam waves specified in the code 2 : downflooding angle or 50 degrees or angle of second intercept between lw2 and GZ curves, whichever is less.

GZ

b

lw2

lw1 a

2 0

Angle of heel

1

Fig. 1 Weather criterion

In the revised code, lw1 and 1 can be evaluated by model experiments in the following conditions and it is allowed to consider the heeling lever as dependent on the heeling angle like the broken line in Fig. 1, lw1 : to the satisfaction of the Administration 1 : when the parameters of the ship are out of the following limits or to the satisfaction of the Administration;

Evaluation of the Weather Criterion 67

- B/d smaller than 3.5 - (KG/d-1) between -0.3 and 0.5 - T smaller than 20 seconds where B, d and KG are the breadth, draft and the height of CG above keel of the ship respectively, and T is the natural rolling period.

3 The Subject Ship The subject ship is a Japanese Ro-Pax ferry. Table 1, Figs. 2 and 3 show the principal particulars, the general arrangement and GZ curve respectively. Compared to general European Ro-Pax ferries this ship has finer shape. From Fig. 3 it is found that up to about 40 degrees the GZ curve has a very small nonlinearity to heel angle. Table 1 Principal particulars

Length between perpendiculars Breadth Depth Draft Displacement Blockage coefficient Area of bilge keels Vertical center of gravity Metacentric height Flooding angle Natural rolling period Lateral projected area Height of center of AL above WL

Lpp B D D W Cb B/d Abk KG G0M

f

Tr AL Hc

Fig. 2 General arrangement

[m] [m] [m] [m] [ton] [m2] [m] [m] [deg] [sec] [m2] [m]

170.0 25.0 14.8 6.6 14,983 0.521 3.79 61.32 10.63 1.41 39.5 17.90 3433.0 9.71

68 S. Ishida et al.

1.4 1.2 GZ [m]

1 0.8 0.6 0.4 0.2 0 0

20

40

60

80

Angle of Heel [deg]

Fig. 3 GZ curve

4 Wind Heeling Lever lW1 The wind heeling lever is estimated from the heeling moment when the ship is drifting laterally by beam wind. Therefore, wind tunnel tests and drifting tests are necessary.

4.1 Wind Tunnel Tests The wind tunnel tests were conducted at “Pulsating wind tunnel with water channel” of NMRI (National Maritime Research Institute), with wind section of 3m in width and 2m in height. The test arrangement is shown in Fig. 4. The connection between the model and load cell had a rotating device for testing the model in heeled conditions. In heeled conditions, the height of the model was adjusted by the adjusting plate to keep the displacement constant when floating freely. By using the model of 1.5m in length, the blockage ratio was kept less than 5%, which is requested by the guidelines. The gap between the model and the floor plate was kept within approximately 3mm and covered by soft sheets for avoiding the effect of downflow through the gap.

Evaluation of the Weather Criterion 69

Fig. 4 Arrangement for wind tunnel tests

The wind speed was varied from 5m/s to 15m/s in upright condition and confirmed that the drag coefficient is almost constant in this speed range. For the full tests a wind speed of 10m/s was used, corresponding to the Reynolds’ number of 1.52  105, as defined by the following equation:

Re 

U B



(1)

where U  is the uniform wind speed outside the boundary layer, B is the breadth of the model and  is the kinematic viscosity coefficient of air. The horizontal force Fwind, the heeling moment M and the lift force L were measured by the load cell. The heeling moment M was converted to the one with respect to point O, defined as Mwind in the guidelines, by the following equation:

M wind  M  Fwind l cos   L  l sin 

(2)

where l is the distance from the centre of the load cell to the point O, which is defined as the cross point of the centre line of the ship and waterline in upright condition.

4.2 Results of Wind Tunnel Tests The measured drag coefficient CD, lift coefficient CL and heeling moment coefficient CM are shown in Fig. 5. They are nondimensionalized by the following equations:

 CD   Fwind   1  2  C    L   2  airU AL      L 

(3)

70 S. Ishida et al.

CM  M wind

2  1 2 AL   airU  L pp  2

CD, CL, CM

CD CL CM CM (standard criterion)

(4)

1.25 1 0.75 0.5 0.25 0

-35 -30 -25 -20 -15 -10 -5

0

5

10

15 20

25

30

35

Angle of heel (deg.)

Fig. 5 Results of wind tunnel tests

In the figure the angle of heel is defined as positive when the ship heels to lee side as shown in Fig. 4. The broken line is the heeling moment coefficient calculated from the current weather criterion (IMO, 2002, called standard weather criterion hereafter). It is characteristic in Fig. 5 that at all the quantities (CD, CL and CM) vary significantly with heel angle. As for the heeling moment, it is smaller than the standard criterion and further reduces when the ship heels, especially to lee side. The lift force is not so small and close to the drag force when the heeling angle is 5 degrees (weather side). However, the adjustment of the vertical position of the model is not necessary since the lift force is 0.7% of the displacement of the ship in the assumed wind speed of 26m/s. The result is also shown in Fig. 6 as the height of the centre of wind force above waterline , lwind, by the following equation. It can be observed that the centre of wind force is also a function of heel angle.

lwind  M wind / Fwind

(5)

Evaluation of the Weather Criterion 71 0.150 0.125

lwind (m)

0.100 0.075 Centre of lateral projected area

0.050 0.025

0.000 -35 -30 -25 -20 -15 -10 -5 0

5

10 15 20 25 30 35

Angle of heel (deg.)

Fig. 6 Height of the centre of wind force above waterline (model scale)

Although it is not requested in the guidelines, the effect of encounter angle,  , was investigated. Fig. 7 shows the wind heeling moment coefficient, CM. Here  <90 means following wind. The figure shows that the wind heeling moment is almost at the maximum in beam wind condition (  =90 degrees). This fact supports the assumption of existing regulations. However, for developing performance based, physics based criteria, the information on the effect of encounter angle to heeling moment might be necessary.

0.75

15 (deg.) 30 45 60

CM

0.50

75 90 105

0.25

120 135 150

0.00 -35 -30 -25 -20 -15 -10 -5 0

5

10 15 20 25 30 35

165

Angle of heel [deg.]

Fig. 7 Wind heeling moment coefficient for various encounter angles

72 S. Ishida et al.

4.3 Drifting Tests The detail of drifting tests was reported in the previous paper. Here, the height of the centre of drift force above waterline, lwater, is shown in Fig. 8. lwater was calculated in the same manner as equation (3). The angle of heel is positive when the ship heels to the drift direction. The drift speed (towing speed) was decided, as requested in the guidelines, to make the measured drift force equal to the wind force at the wind speed of 26m/s in ship scale. Because drifting tests were conducted before wind tests, the wind drag coefficient, CD, was assumed to be from 0.5 to 1.1. In upright condition and CD=0.8 the drift speed was 0.195m/s (1.80m/s in ship scale). CD=0.5 CD=0.7 CD=0.9 CD=1.1

0.6 0.4

CD=0.6 CD=0.8 CD=1.0

Iwater / draft

0.2 0.0 −20 −15 −10 −5 0 −0.2 −0.4

5

10

15

20

25

30

35

Angle of Heel (deg.)

Half draft

−0.6

Fig. 8 Height of the centre of drift force above waterline

Fig. 8 shows that the centre of drift force is above half draft (which is assumed in the standard criterion) and is generally above the waterline for this ship. This phenomenon was reported by Hishida and Tomi (1960), Ishida (1993), Ishida and Fujiwara (2000) and referred in IMO/SLF (2003). This is due to the more dominant effect of the bottom pressure distribution than the side pressure when breadth/draft ratio is large. For the cross sections with this proportion, high position of the centre of sway force can be easily found in hydrodynamic tables of Lewis Form. This fact suggests that potential theory would explain this phenomenon. However, the effect of separated flow, e.g. from bilges, was also pointed out (Ishida and Fujiwara, 2000). It was confirmed experimentally in the previous paper that lwater reduces when the draft is enlarged.

Evaluation of the Weather Criterion 73

4.4 Determination of lw1 The heeling moments by wind, Mwind, and by drifting, Mwater, both around point O, were divided by the displacement,  , and the wind heeling lever, l w1, was calculated as a function of heel angle (equation (6)). Figs. 9 and 10 show the results.

M wind  M water

(6)

 Standard Criterion Wind test Drift test Wind + Drift tests

0.15 0.125 0.1 IW1 [m]

0.075 0.05 0.025 0 −0.025 −0.05 −20

−10

0

10

20

30

40

Angle of Heel (deg.)

Fig. 9 Wind heeling lever, lw1, evaluated by the tests

1

0.5 GZ, Iw1 [m]

lw1 

0 −20

−10

0

−0.5

10

20

30

40

50

GZ Standard Criterion Wind + Drift tests

−1 Angle of Heel [deg]

Fig. 10 Wind heeling lever, lw1, compared with the GZ curve

74 S. Ishida et al.

In Fig. 9, the heeling levers due to wind ( M wind  ) and drift motion ( M water  ) are also included. In both figures, lw1 at angles greater than 30 degrees (tested range) is assumed to keep the same value as at 30 degrees as prescribed in the guidelines. Figs. 9 and 10 show that, in the considered case, the wind heeling lever estimated by wind and drift tests is sensibly smaller than that required by the standard weather criterion.

5 Roll Angle 1 The formula of roll angle 1 in the weather criterion implies the maximum amplitude out of 20 to 50 roll cycles in beam irregular waves. And 1 is related to the resonant roll amplitude, 1r , in regular waves, whose height and period are equal to the

significant wave height and mean wave period of the assumed irregular waves (IMO, 2006, Watanabe et al., 1956). The reduction factor is 0.7 (see equation (7)) and this alternative assessment estimates 1r instead of 1 by model experiments.

1  0.71r

(7)

5.1 Direct Measurement Procedure In the guidelines, this procedure is called “Direct measurement procedure” because the resonant roll angle, 1r , is measured directly in waves with the steepness specified in the IS Code and the period equal to the natural roll period. The results of experiments were mentioned in the previous paper. Here, Fig.11 is shown again. In the figure, “s” is the wave steepness, which is tabled in the Code as a function of the natural roll period. For this ship s=0.0383 (1/26.1), but lower steepnesses were also used. Due to the linearity of the GZ curve the amplitudes reach the maxima at the vicinity of the natural roll frequency in all steepnesses. From this result, 1 was decided as 19.3 degrees (=0.7 1r ).

Roll Amplitude / Wave slope

Evaluation of the Weather Criterion 75 7

S=1/60 =1/40

6

=1/26.1 5 4 3 2 1 0 0.6

0.8

1.0

1.2

1.4

Wave frequency / Natural roll frequency

Fig. 11 Roll amplitude in regular waves

5.2 Alternative Procedures In the guidelines, alternatives procedures are included, i.e. “Three steps procedure” and “Parameter identification technique (PIT)”. In the “Three steps procedure”, the roll damping coefficient N is estimated by roll decay tests. And, the effective wave slope coefficient r is estimated by roll motion tests in waves with smaller value of s. Finally, 1r (degrees) is calculated by the following equation:

1r 

90 rs N 1r 

(8)

This method was adopted when the standard weather criterion was developed and is based on linear theory except roll damping. The previous paper showed that the estimated value of 1 is 19.5 degrees when the resonant roll amplitude at s=1/60 is used. This value of 1 is very close to that by the “Direct measurement procedure” due to the linear feature of GZ curve. The PIT is a methodology to determine the parameters included in the equation of roll motion. Once all the parameters are decided by test data at small wave steepness, the roll amplitude at prescribed s can be extrapolated. In the guidelines, an equation with 9 parameters is presented, in which nonlinear features of roll damping, GZ curve and wave exciting moment are included. PIT analysis is not carried out in this paper. However, the difference of 1 by PIT from other 2 procedures is expected to be limited because of the linearity of GZ curve.

76 S. Ishida et al.

6 Alternative Assessment of the Weather Criterion In the guidelines, simplified procedures on the wind heeling lever, lw1, are also mentioned for making the assessment practically easier. For wind tunnel tests, the lateral horizontal force Fwind and the heeling moment Mwind can be obtained for the upright condition only and considered as constants (not depending on heeling angle). Instead of drifting tests, the heeling moment Mwater due to drift can be considered as given by a force equal and opposite direction to Fwind acting at a depth of half draft in upright condition, as assumed in the standard criterion. And the combinations of complete procedures and simplified procedures are to the satisfaction of the Administration. The comparison of the assessments of the weather criterion using experimental results is summarized in Table 2. In the table all the possible combinations of the wind tests and the drifting tests, complete procedures and simplified procedures are included. As for 1 , the standard criterion and the result of “Direct measurement procedure” are included. The results of “Three steps procedure” can be omitted here since the estimated 1 was almost equal to the one of “Direct measurement procedure” for this ship. Table 2 Assessments of weather criterion by experiments

The last line of Table 2 shows the critical values of the vertical centre of gravity (KG), in which the ratio of area b/a=1 (see Fig. 1). These last results are to be taken with some caution, since the effects of changing the vertical centre of gravity on Tr (natural roll period) and on the other quantities related to roll motion, including 1 , have been neglected. Table 2 shows that the alternative assessment by model experiments can change b/a significantly with respect to the standard criterion. For this ship, b/a’s of the right side of the table are smaller than those of the left side. This tendency comes from the increased 1 obtained by experiments and it was suggested in the previous

Evaluation of the Weather Criterion 77

paper that 1 was enlarged by the small damping coefficient (N=0.011 at 20 degrees). On the other hand, the lw1 evaluated through all the combinations of the wind tests and drifting tests, complete procedures and simplified procedures, tends to make b/a larger than the standard criterion. It has to be noted that the leading cause of the fluctuations is the large variation in the vertical centre of hydrodynamic pressure when evaluated through the drifting tests. Fig. 12 shows the critical values of KG. From Table 2 and Fig. 12, it is recognized that the changes in the critical value of the vertical centre of gravity are more contained than b/a, but as much as 0.4m at the maximum from the standard criterion.

Fig. 12 Variation of critical KG

7 Conclusions In this paper, the results of the alternative assessment of the weather criterion by model experiments are reported. Almost full tests, included in the interim guidelines, were conducted. Considerable fluctuations of the assessment and their reasons are clarified. In order to make the assessment more uniform and to remove the word of “interim” from the guidelines, more extensive confirmation from the experience gained through the application of the guidelines is needed.

Acknowledments Some parts of this investigation were carried out as a research activity of the SPL research panel of the Japan Ship Technology Research Association in the fiscal year of 2005, funded by the Nippon Foundation. The authors express their sincere gratitude to the both organizations.

78 S. Ishida et al.

References Bulian, G, Francescutto, A, Serra, A and Umeda, N (2004) The Development of a Standardized Experimental Approach to the Assessment of Ship Stability in the Frame of Weather Criterion. Proc. 7th Int Ship Stab Workshop 118-126 Francescutto, A, Umeda, N, Serra, A, Bulian, G and Paroka, D (2004) Experiment-Supported Weather Criterion and its Design Impact on Large Passenger Ships. Proc. 2nd Int Marit Conf on Des for Saf 103-113 Hishida, T and Tomi, T (1960) Wind Moment Acting on a Ship among Regular Waves. J of the Soc of Nav Archit of Japan 108:125-133 (in Japanese) IMO (2002) Code on Intact Stability for All Types of Ships Covered by Instruments. Int Marit Organ IMO (2003) SLF46/6/14 Direct Estimation of Coefficients in the Weather Criterion: submitt by Japan IMO (2006) SLF49/5 Revised Intact Stability Code prepared by the Intersessional Correspondence Group (Part of the Correspondence Group’s report): submitt by Germany Ishida, S (1993) Model Experiment on the Mechanism of Capsizing of a Small Ship in Beam Seas (Part 2 On the Nonlinearity of Sway Damping and its Lever). J of the Soc of Nav Archit of Japan 174: 163170 (in Japanese) Ishida, S and Fujiwara, T (2000) On the Capsizing Mechanism of Small Craft in Beam Breaking Waves. Proc. 7th Int Conf on Stab of Ships and Ocean Veh (STAB2000) B:868-877 Taguchi, H, Ishida, S and Sawada, H (2005) A trial experiment on the IMO draft guidelines for alternative assessment of the weather criterion. Proc. 8th Int Ship Stab Workshop Watanabe, Y, Kato, H, Inoue, S et al (1956) A Proposed Standard of Stability for Passenger Ships (Part III: Ocean-going and Coasting Ships). J of Soc of Nav Archit of Japanese 99:29-46

Evolution of Analysis and Standardization of Ship Stability: Problems and Perspectives Yury Nechaev St.Petersburg State Marine Technical University, Russia

Abstract

This paper describes the research and development in the field of ship stability in waves, including works from more than 50 years of the author’s experience. The first works of the author on stability were submitted to IMO in 1965, regarding development of stability regulations for fishing vessels.

1 Introduction Analysis of the behavior of a ship under action of external forces in various operational conditions is one of the most complex problems of dynamics of nonlinear systems. Extreme situations present an especially difficult challenge. All of the mathematical models describing the behavior of a ship in waves have a similar structure; all of them are nonlinear. Weak nonlinearity is a relatively wellstudied area of a ship dynamics. General methods for obtaining solutions and effective algorithms for practical applications exist. The methods for the analysis of ship stability in waves are based on more complex mathematical models and advanced computational methods, due to the nonlinearity of these problems. The analysis of nonlinearity allows insight into the problem and the ability to construct a general theory describing ship behavior as a nonlinear dynamical system for the control and forecasting of stability changes in various conditions of operation. The in-depth research of ship dynamics in waves requires understanding of physical laws, effects, and phenomena.

2 Phases of Stability Research The research of ship stability in waves is complemented by the development of intelligent decision support systems. More frequently these applications use artificial intelligence. These technologies are focused on the process chain: M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_5, © Springer Science+Business Media B.V. 2011

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modeling-forecasting-decision-making using complex integrated intelligent systems (IS). The analysis of “informational snapshots” of the investigated situations allows one to formulate the basic principles determining “latent” information processes. Currently new approaches for intelligent technologies, including non-algorithmic control which is naturally parallel and non-deterministic, are under development. The period of development of the studies about ship stability is marked by a change of paradigms as shown in Fig. 1. Theoretical-experimental paradigm marks the beginning of research on ship stability in waves. At this stage, the basis of experimental and theoretical methods occur within the framework А. N. Кrylov’s hypothesis- the ship does not affect the wave (Lugovsky 1966). The main attention here is given to the development of theoretical models for the description of the righting moment and estimation of dynamic roll of a ship in waves (Boroday and Netsvetaev 1969, 1982). Physical pictures of roll and capsizing ships from experimental research with radio-controlled models of ships with natural waves were also used (Nechaev 1978, 1989). The basic scientific result of the research was the determination of critical situations for a ship in waves (complete loss of stability, low-frequency resonance, broaching) and justification of the necessity for theoretical models of dynamics to account for nonlinear wave interference components — for ships with small L/B ratios having high Froude numbers. The generalization of the results using physical models allowed the author to develop linear regression formula (submitted to IMO in 1965 (Nechaev 1965)), describing the function of the righting moment for a preliminary estimation of stability in waves. The main works of Russian researchers on this paradigm, were performed within 1960-1970. The theoretical-experimental paradigm was implemented on the basis of digital and analog computers with serial processing. Computer paradigm, where the majority of works on ship dynamics in waves is currently carried out in Russia, is connected to the further development of stability theory using methods and models of research of nonlinear dynamical systems (ex. Fokker-Plank-Коlmogorov equation, method of Monte-Carlo, method of moments, phase plane, catastrophe theory), and also using modern methods of data processing of physical experimental data with fast algorithms. Most of the principle advancements in ship dynamics in waves were achieved within the frames of this paradigm (works performed during 1971-1995). Development of software and analysis procedures for ship stability in waves was the most important part of these works; with the particular emphasis on probabilistic models and criteria (Boroday et al. 1989, Nechaev 1978, 1989). Availability of high-performance computers allowed the author to carry out robust analysis of experimental results and develop a nonlinear regression model describing the spatial function of the righting moment on waves (Nechaev 1978, 1989). Different types of bifurcations, including strange attractors and the phenomenon of deterministic chaos were also studied within the framework of the computer paradigm; leading to a better understanding of the behavior of nonlinear dynamical systems describing large-amplitude motions of a ship in waves. Implementation of

Evolution of Analysis and Standardization of Ship Stability 81

the computer paradigm is related with processing of information using parallel and network computing.

Fig. 1 The approaches to paradigms for information processes for research of ship stability in waves

The phenomenon of artificial intelligence; its application in ship dynamics marked a new paradigm. Artificial intelligence is a principle component of onboard intelligent systems (IS) with intended applications for prediction and control of ship motions in waves as well as monitoring of safe navigation. Development of these systems involves significant difficulties related to formalization of knowledge and very complex computation algorithms. This concept considers stability in waves as a complex, open set of coordinated dynamical procedures that are carried out for analysis of extreme situations with an objective of safe navigation. Implementation of new technology for ship control and the prediction of stability on waves requires extensive use of mathematical modeling and supercomputers (Nechaev et al. 2002, Nechaev and Zavialova 2002). The development of new generation IS with a dynamic knowledge base and a system of adaptation requires reconsideration of general principles of organization and formalization of knowledge of interaction between a ship and its environment, as well as with hardware and software (Nechaev and Petrov 2005). The requirements of self-training and self-organizing in a continuously changing and fuzzy environment

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lead to re-examination of the contents of adaptive components of the knowledge base. These requirements also have increased a role of mathematical modeling for prediction of ship behavior in waves as a complex dynamic object (DO). Most of the information here comes from sensors of a measuring system; analysis, prediction and interpretation of extreme situations is carried out on the basis of this information. As a result, problems of integration of information and an analysis of alternatives were revealed. Complimenting the knowledge base with solvers for numerical, logical, and combinatorics problems, enabled the author to achieve significant progress in one of the most complex areas of knowledge engineering – development of methods for processing of information in dynamic environment. The development of new methods of control of information processing gave birth to new generation onboard IS for motion and stability control. The associative approach of this technology allows organizing the process of making decisions for control of a dynamical object by integration of information and algorithms, taking into account restrictions and under-determinate of models. This new paradigm of the control and prediction of ship stability in waves with on-board IS is focused on extreme situations while using a wide variety of mathematical models (Nechaev et al. 2001, 2006). The model of an extreme situation is based on direct interaction with the environment; in this context the application of parallel computing is both necessary and natural. This new type of control is based on the measurements from sensors and results of simulation; such an approach considerably changes organization of data processing by the onboard IS, making it distributed and independent of number of processors. A new conceptual approach is underway to substitute non-random and consecutive computational processes; this process is based on models of the current situation and associative self-organizing in parallel computational processes. Thus, the research and development in the area of ship stability in waves allows consideration of a model of ship/environment interaction as a complex holistic open system. One of the main focuses of the analysis is the development of integrated on-board IS, using models based on fuzzy logic and neural networks. It allows development of more flexible systems and adequate description of a phenomenon of ship behavior as complex dynamic object in conditions of uncertainty and incompleteness of the initial information.

3 Analysis and Criteria of Stability of Ship in Waves. Background Theory and Validation Dynamics of the interaction of a ship with environment as a problem of stability in waves generally can be described as follows:

Evolution of Analysis and Standardization of Ship Stability 83

dx  f ( X , Y , t ), x(t0 )  X 0 , dt F ( X , t )  0, t  [t0 , T ]

(1)

where Х is a n-dimensional vector of phase coordinates; Y is a m-dimensional vector stochastic excitations; F(X, t) is the area of phase coordinates corresponding to a safe operation; f ( X , Y , t ) is a vector-valued function describing the dynamical system and x(t0 )  X 0 are initial conditions. The solution of a problem (1) results in allowable values of output, used for formulation of stability criteria:

R1  Q j ,

j  1,…, J ;

Q1  1 | z1*  FG1  z *j* , "

(2)

Q1   J | z  FG1  z , * 1

** j

KGCR  KGmin . Here 1 , … ,  J are the area of possible values of criteria, taking into account uncertainty and incompleteness of the initial information; KGCR — is the critical elevation of a centre of gravity of a ship. Using a concept of KGCR, it is possible to associate probability of capsizing with a probability of a parameter exceeding some allowable value

P Z G  Z CR   P C1 ,…, Ck  dC1 ,…, dCk



(3)



where  is the area where variables C1 , … , Ck satisfy the following inequality

Z G (C1 , … , Ck )  Z CR

(4)

An alternative formulation of a problem is that the value of KGCR can be exceeded with the given probability P0 :

R( Z G  Z CR )  P0

(5)

Therefore, the problem formulation (1), (2) consists of developing an algorithm for the analysis and judgment of stability in waves with an estimate of correctness

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of formulation of criteria in the conditions of uncertainty and incompleteness of the initial information. The following theoretical principles are formulated for the research of ship stability in waves and further development of stability standards. Consider an information operator processing information on dynamics of interaction of a ship with environment. A structure of this operator should reflect the following properties of information flows (Nechaev 1995):  Relatively small uncertainty in input may be related with significant uncertainty of the output. This uncertainty must be accounted correctly, as the output represents vital stability information.  Formulation of stability criteria should take into account properties of a ship as dynamical system under external excitation. A normal operating area is around stable equilibria with a corresponding safe basin parameter space. The behavior of system on and in vicinity of the boundaries depends on the physics of the problem. Uncertainty leads to appearance of a “gray” area limited by “safe” and “unsafe” boundaries. Small excursions through the “safe” boundary lead to equally small changes in the state of the dynamical system, while similar excursions through the “unsafe” boundary lead to a qualitatively different state of the system that is incompatible with safe operation. Practical implementation of these theoretical principles for setting boundary values for criteria results in difficulties caused by the uncertainty and incompleteness of the input data. The generalized model developed within the framework of this approach allows formal consideration of ship stability in waves, including both analysis and criteria. Research of ship stability in waves includes consideration of direct and inverse problems. The solution of a direct problem is the transition from a known structure and inputs of a dynamical system to the characteristics of output and stability criteria. The objective of the inverse problem is synthesis; it is transition from the desirable characteristics and known criteria to unknown structure of the dynamical system and characteristics of its components. These models characterize the theoretical formalized core of stability regulations. Formulation of the problems of the control and prediction of stability in waves is inherently related with conditions of uncertainty and incompleteness of the input data. Therefore, the problem of validity of the mathematical models becomes the most important. Let Fi (a), (i  1, … , n) be a criterion of validity defined from the analysis of a mathematical model describing a certain type (or types) of ship motions. This criterion is either a function of parameters a j , ( j  1, … , k ) , or a function of the solution of differential equations. The coefficients of mathematical model a j , satisfying given parametrical, functional and criteria restrictions, make an allowable area Ea in a space of criteria F ( Ea ) . Setting the accuracy of approximation of parameters a j as  j , ( j  1, … , k ) , and the accuracy of criteria

Evolution of Analysis and Standardization of Ship Stability 85

 i , (i  1, … , n) , it is possible to present criterion for the proximity of the data from mathematical modeling to data from physical experiments, which is the criterion of validity of the mathematical model:



F  F1P  F1E , … , FnP  FnE



(6)

where index P stands for calculation and E for experiment. The approximation of area Ea is necessary for the evaluation of coefficients of a mathematical model with the given accuracy. It is achieved by finding values such as:

min F (a)  F (a r ), a  F (a ), r  1, … , N

(7)

under conditions for determining allowable area Ea

Fi P  Fi E  Fi**

(8)

Here a1 parameters that are within the boundaries a*j  a j  a*j* , N – number of experiments; Fi** - allowable criteria limits (level of validity), assigned with account for accuracy of the experiment. Attempt to make dependences (6)-(8) specific, leads to the following formulation. Let's present mathematical model (1) as local discrete transformation:

Si   m (Wi , Si1 , Vi ) ,

(9)

where Si is the modeled state of a ship in the i-th instant of time; while Si 1 is the state corresponding to the previous instant of time i – 1; Wi and Vi are variables discriminating state of environment and internal state of the dynamical system in the i-th instant of time. The condition of validity is determined as

Y S

 c  Cm ,

(10)

where S is a domain limiting meaningful behavior of model; Cm is a set of objectives for modeling. For example, if a model of ship motions is linear and its domain of meaningful behavior is limited to relatively small-amplitude motions, it is valid if the objective is a conventional seakeeping analysis. Development of the model, indeed, includes a choice of the type of the model and its domain. To quantify deviation from a penalty function is introduced:

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 0 if Y  S .  Y , S     Y , S  if Y  S

(11)

The discussion at IMO during 1965-67 was focused on the problem of validity of mathematical models of stability and the correct design of physical experiments. Observed differences between the theoretical and experimental data may be used as criteria of validity with the peak data of motion time history (deterministic or probabilistic expressions for linear or angular motions, velocities and accelerations), as well as the limitation of divergence between time histories of phase trajectories. Alternatively, the applicability of theoretical distributions (Gaussian, Rayleigh, Weibull, etc) can be tested against the empirical data. The improvement of mathematical models of ship stability in waves can be facilitated by improvement of the theoretical background and experimental technology of hydro-aerodynamic research, the use of effective methods for processing and analyzing experimental results. The validity of mathematical models can be improved by the development of more reliable ways of evaluation of components of these models by better accounting for distortions that a moving vessel introduces into the wave field (interference and diffraction of waves, change of a field of pressure), as well as a more complete description of the spatial and temporary structure of wind air-flows and roll damping forces. Mathematical models for stability in waves (Boroday and Netsvetaev 1969, 1982, Lugovsky 1966, 1971, 1980, Nechaev 1978, 1989) are usually developed for extreme situations. The background of these models is a description of the ship interaction with environment, so it is a nonlinear dynamic system with six degrees of freedom (Кrylov , 1958). A specific type of mathematical model is determined by the physics which are observable during experiments. For example, when considering stability in following waves or stability in breaking waves it is enough to use the differential equations of drift and roll. While considering broaching, it is necessary to use a system of at least four differential equations, including the surge, sway, roll and yaw. In more complex situations, it is necessary to formulate hypotheses with all six degrees of freedom included. It is important to include a term describing continually changing stability in waves in the equation of roll motions. The nonlinear spatial function of the righting moment in waves is represented by the following formula (Nechaev, 1978,1989):

M W  M ( ,  , t )  D[l ( ,  )   l ( ,  ) cos( k t   )]; l ( ,  )  0.5[l ( ,  ) max  l ( ,  ) min ],  l ( ,  )  0.5[ l ( ,  ) max   l ( ,  ) min ]; (12) M W   ( ,  k , t )  D l ( ,  , t ), where  l ( ,  ) max and  l ( ,  ) min are the magnitude of stability changes while the ship is on a wave crest or a wave trough, at various course angles ï ; l ( ,  ) is

Evolution of Analysis and Standardization of Ship Stability 87

the righting arm determined by interpolation on  and  for the various moments of time,  is the phase of a ship relative to a wave. 0,3

M( , ,t)/D 1

2

0.3

3

0,2

,t)/D M( ,

2



1

0.2

0,1

0.1

0

0 -0,1 0



80



60

30

40



o

-0.1 0



20

M( , ,t)/D 3

40

20

0

90

60

80

o



M( , ,t)/D

2

1

60

k

o



0

0.3

1

0,2

2



0.2

0 0.1 0

-0,2

80

60 o



40

20

0

0







o



-0.1 0 30

k

60 90

0

20

40

60

80

o



Fig. 2 Nonlinear function, describe of righting component: 1 – initial function; 2 – transform function; 3 – wave shape

The geometric interpretation of function M ( ,  , t ) is given in Fig. 2.

4 Methods for the Analysis of Nonlinear Dynamic Systems in the Modeling of Ship Stability in Waves The research on ship stability in waves as a complex nonlinear dynamic system requires understanding of the physics governing the phenomena. These problems not always can be resolved within the frames of existing concepts and approaches and may require development of the new approaches, methods and models. Nonlinearity is, probably, one of the most important properties of the system, which defines the methods applicable for analysis, such as: method of MonteCarlo, method of the moments, method of action functional, equation of FokkerPlank-Кhоlmogorov, method of a phase plane and catastrophe theory. These methods are helpful in both the qualitative and quantitative interpretation of complex physical phenomena associated with stability of ship in waves. The author applied the Monte-Carlo method for stability and roll of a ship in waves in 1975; it is described in the monograph (Nechaev, 1978). This method plays a role of a bridge between theory and experiment. Being quite a powerful tool of research in statistical dynamics, the Monte-Carlo method is related with handling rather extensive volumes of data. It requires high speed of computation

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and a large volume of memory. For the solution of such problems, high-performance computers with parallel data processing technologies are best. The mathematical model for the application of the Monte-Carlo method is transformed into a dynamical system containing stochastic nonlinear function representing the righting moment. This function can be presented as

M ( ,  , t )  {1   (t ) cos[k t   0 (t )]}(2  a  2 sgn  ). Jx   Jx

(13)

Parameter (t) and phase (t) are considered random variables with normal and uniform distribution, respectively.

 (t )  {M *[  (t )], D*[  (t )]},  0 (t )  [0, 2 ]

(14)

The hypothesis of Gaussian distribution of the process (t) is checked by the analysis of the time history of process. The correlation function R* (t ) of the process (t) is estimated using the data of real sea waves. Calculations show, that the function R* (t ) can be presented with the following approximation:

R* ( )  D *[  (t )] exp[ (  ) | t   |] cos  (  )(t   ),

(15)

where () and () — parameters to be found by the calculations. For modeling the stochastic process (t), the method of forming filters is used in time domain. The parameters of the filter can be found from variance D *  (t ) and autocorrelation function R* ( ) . The filter is described by system of the differential equations:

d    (  )  (t )   (  ) (t )  2 (  ) D  * W1 (t ); dt d    (  )  (t )   (  ) (t )  2 (  ) D * W2 (t ), dt

(16)

where M *[W1 (t )W1 ( )  * (t   ); M *[W1 (t )W2 ( ) 0; M *[W2 (t )W2 ( )  * (t   ); M *[ (t ) 0, M *[  2 (0) D  * ; M *[ 2 (0) D * , D   D ; M * [  (0) (0) 0.

(17)

Evolution of Analysis and Standardization of Ship Stability 89

Here W1 (t ) and W2 (t ) are white noise stochastic processes; () and () are parameters of the autocorrelation function; (t) is the auxiliary stochastic process; M * and D* are operators of mean and variance;  * is the delta function. This mathematical model is a part of practical procedure of modeling the system with the given initial conditions. An important point here is the necessity of formation of the random initial conditions determined by expressions (17). As shown by calculations, the Monte-Carlo method has a number of advantages over other methods of solution (method statistical linearization, theory of Markov processes) for problems of statistical dynamics of nonlinear systems. The main advantages are compactness of the calculation scheme, stability of results in a case of hardware failure, and a rather simple way to estimate the accuracy of results. Another convenient feature is that there is no limitation on the structure of differential equations as any type on nonlinear terms can be included. A traditional approach of classical dynamics considers either deterministic (regular) or stochastic (irregular) processes described by the appropriate nonlinear differential equations of roll motions. However, the nonlinear deterministic dynamical system can exhibit chaotic behavior, when phase trajectories that were initially close, become divergent in the limited area of phase space. This phenomenon of the nonlinear system is known as deterministic chaos. Nonlinear dynamics makes a clear distinction between simple (usual) and strange (chaotic) attractors. The simple attractors are encountered quite frequently when describing nonlinear roll and capsizing of a ship on waves. The geometrical interpretation of a simple attractor describing a dissipative system represents on a phase plane either a focus or a limit cycle. For both these cases, all phase trajectories have a shape of spirals that converge to a stable equilibrium or to steady-state stable cycle. Strange attractors have more complex structure, as they contain both stable (attractors) and unstable (repeller) trajectories. Essentially, they are saddletype trajectories that are being attractors in one dimension and repellers in another; they form a set of layers connecting to each other in quite complex ways.

Fig. 3 Formation of a limiting cycle caused by of group waves

The papers (Degtyarev and Boukhanovsky 1995, 1996) describe the development of parametric resonant motions of a ship caused by passage of group of waves. It is shown that complex structures of oscillatory roll modes are formed in this case. These structures are sets of attractors looking like unstable limit cycles. The cycleattractor can lose stability through several scenarios. A passing group of large

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waves stabilizes the limit cycle described due to the nonlinearity of large-amplitude response (Fig. 3). This cycle appears when the wave height in a sequence of waves in a group exceeds the critical value of creating enough parametric excitation to reach practically constant amplitude of response. However, the subsequent gradual reduction of wave heights breakes the conditions for parametric resonance and the cycle disappears. A more complex scenario is realized when the stable cycle collides with an unstable cycle (Fig. 4). Such a situation is much rarer and it can be generated by a consecutive passage of waves groups containing wave of significantly different height. The first wave group with small heights results in a relatively smallamplitude limit cycle, while the second group generates large-amplitude cycle. Appearance and loss of stability of an oscillatory mode (“birth and death of a cycle” using a terminology by А.А. Аndronov) occur due to limitedness of a rolling resonant zone on a rather small time interval where large-amplitude motions are generated by a passage of wave groups.

Fig. 4 Appearance (A) and loss of stability of a cycle (B) (C)

The problems and methods of controlled chaos is an area of intensive research in recent decades. In the beginning, the concept of deterministic chaos was considered as an exotic phenomenon that is of interest to mathematicians only; the very possibility of occurrence of chaotic response in practical applications was doubtful. However, later on, chaotic dynamics was found to be present in many dynamical systems across disciplines: in mechanics, laser physics and radiophysics, chemistry and biology, economy, and health science. Systems with chaos demonstrate both good controllability and surprising flexibility: the system quickly reacts to external excitations, while keeping a mode of motions. The combination of controllability and flexibility in part is the reason why chaotic dynamics is a characteristic type of behavior for many dynamical systems describing ship motions in waves. From the point of view of the synergetic approach, chaotic systems represent possibilities for realization of self-organizing processes. One of the most typical scenarios for the transition to chaos is through a sequence of period-doubling bifurcations, which is observed for systems with viscous friction under action of excitation forces. The research of chaotic behavior of a nonlinear dynamical system describing roll motions was carried out by systematic variation of control parameters. The result was shown on Poincare map, allowing to clearly see the

Evolution of Analysis and Standardization of Ship Stability 91

period-doubling and sub-harmonic bifurcation (Nechaev 1993). Another scenario is the mechanism of transition to chaotic response, known as intermittency, where intervals of deterministic chaos alternate with almost periodic motions (Fig. 5).

t

1

2

A

B

Fig. 5 Occurrence of chaos of an alternated type: А – Duffing model; IB – Mathieu model at presence of an initial heel angle.

The Lyapunov exponent is an important tool for study of behavior of chaotic systems. It can be demonstrated with the Duffing equation, which is one of the interpretations of nonlinear rolling (Nechaev, 2007). Introduction of the third dimension for the independent variable allows presentation of the Duffing equation as a system of the third order: x2   f1 ( x)      3 f ( x)   f 2 ( x)     x1  k x2  B cos x3    f ( x)   1  3   

(18)

This system is defined with two positive parameters: coefficient of linear damping k and amplitude of excitation B. Negative divergence is determined by damping (dissipation)

 f1  f 2  f 3    k  0  x1  x2  x3

(19)

The linearized system leads to the following matrix

 0  A    3x12  0 

1 k 0

0    B sin x3   0 

(20)

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The results of calculations for В = 10, k =0.1 are shown in a Fig. 6. These parameters lead to the appearance of chaotic behavior of the system (18), which is confirmed by time histories, phase trajectories and the Lyapunov exponent. Theoretical development of chaotic dynamics has revealed a series of possible practical applications, including ship motions and stability in waves, where the chaotic responses may be encountered as a result of complex interaction of the ship and environment. Moreover, there are possible practical applications, where the control over nonlinear system is realized by changing a degree of its “chaoticness”. Methods for the solution of similar problems have been developed recently, especially for application of different control algorithms for chaotic systems.

Fig. 6 Attractor and Lyapunov exponents for the Duffing equation

5 Analysis of Extreme Situations and Control of Ship Stability in Waves The evaluation and prediction of stability changes for motion control of a ship in waves are carried out using the information from acting sensors (Nechaev (ed.) 2001). In this case, the extreme situation caused by deterioration of stability can be described in compact set with variable z  (1, … , m) . The analysis produces scenarios for the possible development of extreme situations, taking into account physics of the interaction between a ship and environment. When the probabilities of encounter of extreme situations are estimated, measures of statistical uncertainty have to be evaluated as well. Let number of situations be N, while M(w) is an average

Evolution of Analysis and Standardization of Ship Stability 93

estimate, and D(w) is the variance estimate of a value representing possible consequences of the situation.

N 0j  q j N 0

( j  1, … , m)

(21)

where N 0 is the predicted total number of extreme situations for the considered period of time; q j is the fraction of situations for the j-th class in the distribution F(w), determined by the formula

q j  P ( wmin j  w  wmax j )

(22)

The value q j is calculated with known distribution F(w) and its parameters M(w) and D(w). The accuracy of the forecast under the formula (21) is characterized by the standard deviation:



 [ N10 ]  q j [ N 0 ]  N 0 [q j ] 





(23)

2 1/ 2

2



2





2

random deviation q j [N 0 ] and N 0 [q j ] included in the formula (23) are determined by statistical processing of results from the mathematical modeling. The development of an algorithm for the extreme situation prediction is carried out using the information from a measuring system. This information is preprocessed and represents a time-series reconstructed on the basis of the Takens’ theorem. According to the Takens’ theorem, a phase portrait restored as:

Z (t )  [ y (t )]  [ y (t ), y (t   ), … , ] y[t  (m  1) ]  [ z1 (t ), …, zm (t )]

(24)

is equivalent topologically to the attractor of initial dynamic system.

6 Method of Action Functional The search of a universal principle that could help to describe behavior of a dynamic system, has resulted in discovery of a principle “of least action”, similar to a principle of “minimum potential energy” that determines the position of equilibria. According to a principle of the least action, the variation problem is a search of a stationary value of the action integral, defined on the interval (t1 , t2 ) . Responses of the dynamical system corresponding to all possible initial conditions are compared.

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If the interval of time (t2  t1 ) is small enough, the action integral has a minimal value as well as the stationary value. The implementation of the algorithm of functional action for ship dynamics in waves is considered in (Nechaev and Dubovik 1995). The advantages of the method can be summarized as follows:

 use of the information for an external disturbance, received during real-time measurements;  taking into account significant nonlinearity of a problem for any kind of function – possibility of the solution without use of a priori data on behavior of system and its initial conditions. Functional action is defined by an asymptotic expression (Nechaev (ed.) 2001, Nechaev and Dubovik 1995):





lim  2 ln Px X t  D   min S0T  

  C0T R

r n

(25)

,   x, ,    D 0

t 1

It allows estimating probability Px that critical set D (capsizing) will be achieved. This estimation is carried out using optimum control for the stability criteria formulated above. The solution of such problem (25) is given by the main term of logarithmic asymptotic expansion of the probability at   0 .

уз

Fig. 7 Probability of capsizing as a function of time (А), dynamics of roll of a ship on waves (B) and polar diagram (С)

This approach makes possible the development of new methods for analysis and prediction of behavior of nonlinear dynamical systems in real-time, including stability control in waves. The results of a method of functional action are presented as a probability of capsizing as a function of time (Fig. 7A, В). Then interpretation of the results for the given external conditions allows “drawing a picture” of capsizing for a ship in a considered extreme situation. Additionally information on heeling moment leading to capsizing can be made available on the screen. Using the method of the functional action it is possible to evaluate combinations of dangerous

Evolution of Analysis and Standardization of Ship Stability 95

speeds and headings relative to waves and present those in a form of a polar diagram (see Fig. 7C).

7 Study of Ship Stability Using Climatic Spectra and Scenarios of Extreme Situations Modelling the environment is one of the most complex problems that need to be solved while developing procedures for onboard IS. Contemporary approaches to the solution of these problems require new presentations of wind-wave fields in the ocean. These include the conception of climatic officially accepted on 18th Assembly of IMO in 1993 along with the concept “wave climate” (Boukhanovsky et al. 2000). These concepts open the opportunities for more detailed description of waves in different parts of the World Ocean. The hydrodynamic model of waves in the spectral form is represented as the equation of the balance of wave energy:

N N N  N  N  N     k    G t   k  

(26)

Here N is the spectral density of wave action; it is function of latitude , longitude , wave number k, and the angle between a wave direction and a parallel , as well as frequency  and time t. The equation (26) relates inflow of energy from wind, dissipation and redistribution, as well as nonlinear interaction between frequency components of the process. Often, the source function G is written as a sum of three components G  Gin  Gnl  Gds (incoming energy from wind to waves, weak nonlinear interaction in a wave spectrum, and dissipation of wave energy). Examples of classification of wave spectra are shown in Fig. 8. There are six classes of spectra: A – swell; B – wind waves; C – combined swell and wind waves systems with prevalence of swell; D – combined swell and wind waves systems with prevalence of wind waves; E – combined swell and wind waves systems without clear division and with prevalence of swell; F – combined swell and wind waves systems without clear division and with prevalence of wind waves.

Y. Nechaev

96

А

B

C

D

E

F

Fig. 8 The typical normalized spectra of sea waves; vertical axis is value spectral density S()/Smax and horizontal axis is circular frequency , in sec–1

It is important to note that the standard calculation of dynamics of complex objects in real sea waves is done with a one-peak-spectrum, which may result in a type II error in comparison with climatic spectrum. Therefore, climatic spectra must be used for development of onboard IS, as type II errors may result in catastrophic consequences.

8 Paradoxes of Stability Standards The concepts of complexity and randomness are closely related. Simple criteria are often preferred over the complex ones. However, while gaining better understanding of physics of ship/environment interaction, it becomes clear that the problem of a choice of criteria is a very complex one. That’s why sometimes there are unexpected results and paradoxes (Nechaev, 1997). Paradox of “ideal” norms. It is known that the guarantee (defined as a probability that failure will not occur if an object satisfies the norms, see also (Belenky and Sevastianov 2007, Belenky et al. 2008)) for any, even most primitive, norms can be made very close to unity (ideal norms), if

    2  exp   d  0, 2 y x2  y 2   0

1



 KG  KGCR   100; y  KG  

i  



y0

i   

2

N 1

(27)

, 

1 i N

where KG, KGCR are the actual and critical position of center of gravity.

Evolution of Analysis and Standardization of Ship Stability 97

It can only be achieved by making norms stricter. Simple calculations show that considerably more physically valid stability criteria may result in further deviations from the ideal, compared to simplified IMO criteria. The paradox is that the result is achieved by the extremely uneconomical way, at the expense of making a norm stricter than the average norms, even when it is not really necessary. Paradox of zero probability. The paradox of zero-probability is directly related to probabilistic stability criteria. Practically, the probability of capsizing equals to zero (based on casualty statistics, probability of capsizing is about 10–4, i.e. corresponds to a range of risk (1-10) 10–4 similar to landing helicopters, horse races with obstacles and sports car races), but this event is not impossible. There is a paradox, whether it is possible to compare “chances” of events having zeroprobability. Also whether it is real when the conjunction of events having zeroprobability can result in finite-value probability, i.e. the addition of many “nothings” results in “something”. Probabilistic analysis of stability failures leads to a concept of rare events coming from “unusual occurrences” (combination of external conditions for assumed situations, encounter with extreme waves, etc.). The small probability of such events may make a false impression that simultaneous combination of many adverse factors is practically impossible, like conditions that lead to large-amplitude rolling (synchronous and parametrical resonance). This paradox is also closely related to the problem of rarity. Paradoxes of the distributions. Many paradoxes are related with the application of theoretical distributions. One such peculiarity is caused by asymptotic properties of theoretical distributions and results in insensitivity of the probability of stability failure to the large changes of the centre of gravity in the vicinity P0  1 . Other problems are related to the change of distributions by a nonlinear dynamical system, depending on the level of excitation. It is known that for the normal distribution, the average  *   i / N of a



random variable  is an unbiased estimate

E ( * )  0

and converges

P(|  *   |   )  1  at the large values N. However if the distribution is not known a priori, the estimate  * may end up being biased with the minimum variance; in case of multi-dimensional distributions, such an estimate may simply not exist for quadratic loss function (   * ) 2 . If such a check was not



performed during the statistical analysis the resulting criteria may be not valid. Paradox of a choice of boundaries for a criterion. The paradox of rational choice boundaries for a stability criterion is very important. Use of fuzzy boundaries for stability criteria makes sense due to the random nature and uncertainty of the input data that is used as a basis for calculation scheme of evaluation of stability (Fig. 9) (Nechaev, 1997).

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The errors of an inclining experiment, limited accuracy of data on loading conditions of a ship, as well as the uncertainty of other factors, may result in the true value of the parameter Х actually belonging to the interval [ X 0   ; X 0   ] . Then according to the requirements when X *  X 0 , the norm is observed exactly, while in reality X 0   . The area of acceptance of a hypothesis X  X 0 , i.e. the interval ( X 1 , X  ) equals  ( - significance value). Then the probability of a type II error for deviation from hypothetical value X q equals ; the value 1 –  characterizes “power of the criterion”. Reducing  leads to a reduction in the probability of a type I error (the zero hypothesis is rejected, when it is correct). However, the probability  is increased of the type II error and the “power” of criterion is reduced.

X min

X max

Fig. 9 Errors of practical use of stability criteria: X 0 – parameter; ( X max , X min ) - area of actual change Х; A – area zero-hypothesis (1) and alternative hypotheses (2); B – areas corresponding to a type II error

Paradoxes of computer implementation. Many paradoxes are related to computer implementation of stability problems. Any statistical solution, which can be implemented on the computer, now becomes available. As a result, “stable” and multi-dimensional methods, requiring huge number of operations, have entered engineering practice without a sufficient theoretical justification. Meanwhile, it is possible to justify many empirical collisions, using robust statistical methods. Unfortunately this is a common practice in stability assessment and regulation.

Evolution of Analysis and Standardization of Ship Stability 99

9 Future Directions The further developments of the analysis and criteria of stability can be achieved by advancements in the following strategic directions: 1. Development of hydrodynamic model of the interaction of a ship with external environment while under action of extreme waves. 2. Study of physical response of roll and capsizing of a ship in extreme situations using climatic spectra. 3. Development of a hydrodynamic model for nonlinear interaction of ship waves with incident waves in storm conditions. 4. Further development of criteria for ship stability in waves using new approaches for description of uncertainty in complex dynamic environments. 5. Development of effective means of the control of ship stability in waves using algorithms, which take into account the change of ship dynamics in time.

10 Conclusion The considered problems of the analysis and criteria of ship stability in waves reflects only a small portion of the research applications in which the ideas developed by the author, his students and colleagues have found a place. The foundation for these works combines stability and the theory of oscillators and offers a methodology for better understanding how both these concepts are applied for modeling. Thus the author emphasizes applications that attract both scientific and practical interest while researching complex behavior of ships in various extreme situations. Making decisions for the control of these situations may benefit from the new approaches based on formulations of uncertainty and construction of system of knowledge using artificial intelligence and fuzzy mathematics. Advancements in cybernetics, information technology and the general theory of systems led to a new scientific picture of the world. This picture has found its reflection in new approaches to the problem of stability of a ship in waves. The leading role belongs to nonlinear dynamics and is related to the concept of synergy that has changed the understanding of the relation between chaos and order, entropy and information. The theory of synergy came from works on the theory of bifurcation of dynamical systems and was further developed by new generations of researchers. Complexity of structures and processes plays a key role for modeling of the stability of a ship in waves. Their instability, randomness, and transitive nature led to new scientific paradigm. Changes of the structure of a dynamical system are studied within this paradigm, rather than a behavior of a system with unchangeable structure. The problem of analysis and development of stability criteria is one of important avenues for ensuring safety of operation. Uncertainty and incompleteness of the input

100 Y. Nechaev

information are inherent for the complex problem of ship stability. Information technology for monitoring ship dynamics in waves being developed by the author and his colleagues may be implemented into on-board intelligent systems. Such technology should not be reflected in the final form of criteria, as a mathematical formulation allows more flexible analysis of stability in complex situations.

Acknowledgments The author is grateful to his advisor prof. N.B. Sevastianov, who formulated the general approach to study the very difficult problem of ship stability in waves, and also to professors S.N. Blagoveshchensky, I.К. Boroday, N.N. Rakhmanin, V.V. Semenov-Тyan-Shansky and V.G. Sizov – for their support and advice.

References Ananiev DM (1964) On directional stability of a ship in waves. Trans of Krylov Soc 54. Leningrad (in Russian) Ananiev DM (1981) On stability of forced rolling of vessel with given GZ curve. Trans of Kaliningrad Inst of technol 93 Seakeeping of ships, Kaliningrad (in Russian) Ananiev D (1994) Determination the boundaries of surf-riding domain analyzing surging stability. Proc. of the 5th Int conf STAB-94 5 Melbourne, Florida, USA Belenky V, Sevastianov N (2007) Stability and safety of ships: Risk and capsizing. Second Edition SNAME, Jersey City, NJ. Belenky V, de Kat J, Umeda N (2008) Toward performance-based criteria for intact stability. Marine Tech 45(2):122–123 Blagovescensky SN (1951) On stability regulations of sea vessels. Trans of Central Mar Inst. 8, Leningrad (in Russian) Blagovescensky S (1962) Theory of ship motions. In two volumes, Dover Publ, New York Blagovescensky SN (1965) On the wind load on vessels. Trans of Russian Register of Shipping, Theor and Pract probl of stab and survivability, Transp Publish Leningrad 100–146 (in Russian) Blagovescensky SN, Kholodilin AN (1976) Seakeeping and stabilization of ships on wave. Sudostroenie, Leningrad (in Russian) Boroday IК, Netsvetaev YA (1969) Ship motion on sea waves. Sudosroenie, Leningrad (in Russian) Boroday IК, Netsvetaev YA (1982) Seakeeping. Sudosroenie, Leningrad (in Russian) Boroday IK, Morenshildt VA, Vilensky GV et al. (1989) Applied problems of ship dynamic on waves. Sudostroenie, Leningrad (in Russian) Boukhanovsky A, Degtyarev A, Lopatoukhin L, Rozhkov V (2000) Stable states of wave climate: applications for risk estimation. Proc of the 7th Int Conf STAB’2000, Launceston, Tasmania, Australia 831–846 Degtyarev A (1993) Distribution of nonlinear roll motion. Proc of Int Workshop. On the probl of phys and math modeling. OTRADNOYE-93 2 Paper 9 Degtyarev A, Boukhanovsky A (1995) On the estimation of the ship motion in real sea. Proc of Int symp. Ship safety in seaway: stability, maneuverability, nonlinear approach. Sеvastyanov Symposium), Kaliningrаd, 2, Paper 8

Evolution of Analysis and Standardization of Ship Stability 101 Degtyarev A, Boukhanovsky A (1996) Nonlinear Stochastic Ship Motion Stability in Different Wave Regimes. Trans of 3rd Int conf CRF-96, St.Petersburg, 2: 296-306. Degtyarev A, Boukhanovsky A (2000) Pecularities of motions of ship with low buoyancy on asymmetrical random waves. Proc of the Int conf STAB’2000, Launceston. Tasmania. Australia, 2: 665-679. Nechaev YI (ed) (2001) Intelligence systems in mar res and technol. SMTU, St.-Petersburg (in Russian) Кrylov АN (1958) Selected papers. Publ Аcad of Sci USSR, Мoskow (in Russian) Lugovsky VV (1966) Nonlinear problems of seakeeping of ships. Sudostroenie, Leningrad (in Russian) Lugovsky VV (1971) Theoretical problems background of stability standards of sea. Sudostroenie, Leningrad (in Russian) Lugovsky VV (1980) Hydrodynamics of ship non-linear motions. Sudostroenie, Leningrad (in Russian) Makov YL (2005) Stability. What is this? Sudostroenie St.-Petersburg (in Russian) Nechaev Y (1965) Method for determining the decrease of righting arms of stability on wave crest. IMCO. PFV/11.1965 Nechaev YI (1978) Ship stability on following waves. Sudostroenie, Leningrad (in Russian) Nechaev YI (1989) Stability on wave. Modern tendency. Sudostroenie, Leningrad (in Russian) Nechaev Y (1993) Determined chaos in the phase portrait of ships dynamic in a seaway. Proc of Int Workshop OTRADNOYE-93 2: 143-145 Nechaev Y (1996) Problem of uncertainty in hydrodynamic experiment planning. Proc of Int symp “Marine intelligence technology” CRF-96, St.-Petersburg: 453-457. Nechaev Y (1995) The algorithm a correct estimation on the stability under conditions exploitation. Proc of Int symp. Ship safety in seaway: stability, maneuverability, nonlinear approach. (Sеvastyanov Symposium), Kaliningrаd, 2: Paper 17 Nechaev Y Standardization of stability: Problems and perspectives. Proc of 6th Int conf STAB’97, Varna, Bulgaria 2: 39-45. Nechaev Y, Degtyarev A, Boukhanovsky A (1997) Chaotic dynamics of damaged ship in waves Proc of 6th Int conf STAB’97, Varna, Bulgaria 1: 281-284. Nechaev Y, Degtyarev A, Boukhanovsky A (1997) Analysis of extreme situations and ship dynamics on seaway in intelligence system of ship safety monitoring. Proc of 6th Int conf STAB’97, Varna, Bulgaria 1: 351-359. Nechaev Y, Degtyarev A, Boukhanovsky A (1999) Adaptive forecast in real-time intelligence systems. Proc. of 13th Int conf on hydrod in ship design, Hydronaf-99 and Manoeuvring-99, Gdansk – Ostroda, Poland: 225-235 Nechaev Y (2000) Problems of experiments design and choice of regression structure for estimation of ship stability in exploitation conditions. Proc of Int conf STAB'2000, Launceston. Tasmania, Australia 2: 965-972. Nechaev Y, Degtyarev A (2000) Account of peculiarities of ship’s non-linear dynamics in seaworthiness estimation in real-time intelligence systems. Proc of Int conf STAB'2000, Launceston, Tasmania, Australia 2: 688-701 Nechaev Y, Zavyalova O (2001) Analysis of the dynamic scenes at architecture of the interacting in real-time intelligence systems. Proc оf 3rd Int conf on marine ind MARIND-2001, Varna. Bulgaria 2: 205-209. Nechaev Y, Zavyalova O (2001a) The broaching interpretation in learning intelligence systems. Proc of 14th Int Conf on Hydr in Ship Design, Szcecin-Miedzyzdroje, Poland: 253-263. Nechaev Y, Degtyarev A (2001) Knowledge formalization and adequacy of ships dynamics mathematical models in real time intelligence systems. Proc of 14th Int Conf on Hydr in Ship Design, Szcecin-Miedzyzdroje, Poland: 235-244 Nechaev Y, Degtyarev A, Boukhanovsky A. (2001) Complex situation simulation when testing intelligence system knowledge base. Computational Science – ICCS 2001, Part 1, LNCS. Springer: 453 – 462

102 Y. Nechaev Nechaev Y (2002) Principle of competition at neural network technologies realization in onboard real-time intelligence systems. Proc of 1st Int cong on mech and elect eng and tech «MEET-2002» and 4th Int conf on marine ind «MARIND-2002», 3 Varna, Bulgaria: 51-57 Nechaev Y, Degtyarev A, Kirukhin I, Tikhonov D (2002a) Supercomputer technologies in problems of waves parameters and ship dynamic characteristics definition. Proc of 1st Int cong on mech and elect eng and tech «MEET-2002» and 4th Int conf on marine ind «MARIND-2002», 3 Varna, Bulgaria: 59-64 Nechaev Y, Zavialova O (2002) Analysis and interpretation of extreme situation in the learning and decision support systems on the base of supercomputer. Proc of 1st Int cong on mech and elect eng and tech «MEET-2002» and 4th Int conf on marine ind «MARIND-2002», 3 Varna, Bulgaria: 23-29 Nechaev Y, Zavyalova O (2003) Criteria basis for estimation of capsizing danger in broaching extreme situation for irregular following waves. Proc of 8th Int conf STAB’2003, Madrid. Spain: 25-34 Nechaev Y, Degtyarev A (2003) Scenarios of extreme situations on a ship seaworthiness Proc of 8th Int conf STAB’2003, Madrid. Spain: 1-10 Nechaev Y, Zavyalova O (2003) Manoeuvrability loss in a broaching regime the analysis and control of extreme situation. Proc of 15th Int conf on hydrod in ship design, safety and operation, Gdansk, Poland: 201-208 Nechaev Y, Dubovik S (2003) Probability-asymptotic methods in ship dynamic problem. Proc of 15th Int conf on hydrod in ship design, safety and operation, Gdansk Poland: 187-199 Nechaev Y, Petrov O (2005) Fuzzy knowledge system for estimation of ship seaworthiness in onboard real time intelligence systems. Proc of 16th Int conf on hydrod in ship design, 3rd Int symp on ship manoeuvring, Gdansk – Ostroda Poland: 356 – 366 Nechaev Y, Degtyarev A, Anischenko O (2006) Ships dynamic on wave-breaking condition. Proc of 9th Int conf STAB’2006, Rio de Janeiro Brazil 1: 409 – 417 Nechaev Y, Makov Y (2006) Strategy of ship control under intensive icing condition. Proc of 9th Int conf STAB’2006, Rio de Janeiro Brazil 1: 435 – 446 Nechaev Y (2007) Nonlinear dynamic and computing paradigms for analysis of extreme situation. Proc of Int conf «Leonard Euler and modern science». Russian Academy of Science, St-Petersburg: 385 – 390 (in Russian) Nekrasov VA (1978) Probabilistic problem of seakeeping. Sudostroenie, Leningrad (in Russian) On-board intelligence systems (2006) Radiotechnik, Moskow (in Russian) Rakhmanin NN (1966) On dynamic stability of ship with water on deck. Trans of Krylov Soc. 73 Sudostroenie, Leningrad (IMCO STAB/INF. 27.10.66. Submitted by USSR) (in Russian) Rakhmanin NN (1971) Approximate assessment of safety of small vessel in seas. Trans of Russian Register of Shipping, Leningrad: 12 – 24 (in Russian) Rakhmanin N (1995) Roll damping when deck is submerging into seawater. Proc of Int Symp «Ship Safety in Seaway Stability, manoeuvrability, nonlinear approach» (Sеvastyanov Symp) Kaliningrad 2: Paper 1 Sevastianov NB (1963) On probabilistic approach to stability standards. Trans of Kaliningrad Inst of Techn 18: 3–12 (in Russian) Sevastianov NB (1968) Comparison and evaluation of different systems of stability standards. Sudostroenie 6: 3–5 (in Russian) Sevastianov NB (1970) Stability of fishing vessels. Sudostroenie, Leningrad (in Russian) Sevastianov NB (1977) Mathematical experiment on ship capsizing under combined action wind and waves. Rep of Kaliningrad Inst of Tech, No77-2.1.6 (in Russian) Sevastianov NB (1978) On possibility of practical implementation of probabilistic stability regulations. Sudostroenie 1: 13–17 Sevastianov NB (1993) Theoretical and practical models for probabilistic estimation of vessels stability. Proc of Int Workshop OTRADNOYE-93 1: Paper 5

SOLAS 2009 – Raising the Alarm Dracos Vassalos* Andrzej Jasionowski** *The Ship Stability Research Centre (SSRC), Dept of Naval Architecture and Marine Engineering, University of Strathclyde, UK** Safety at Sea Ltd (SaS), Glasgow, UK

Abstract

In anticipation of the new harmonised probabilistic rules for damage stability being adopted in January 2009, a number of ship owners are opting to follow these rules in advance, somewhat hesitantly and reluctantly considering the general lack of experience and understanding but also the confusion that prevails. In this climate, the authors have found a fertile ground for introducing a methodology to deal optimally with this problem, deriving from the arsenal of tools and knowledge available at SSRC and SaS. As a result, the authors are involved in some way or other with most ships currently being designed in accordance with the new probabilistic rules for damage stability. This involvement has revealed a somewhat more serious problem with probabilistic damage stability calculations, which is hidden in the detail of the rules, but one ht at matters most. The authors are highlights this problem and recommends a way forward.

1 Introduction In January 2009, the new harmonised probabilistic rules for ship subdivision will become mandatory, are rules initiating rule making? in the maritime industry, in line with contemporary developments, understanding and expectations. This will be the culmination of more than 50 years of work, one of the longest gestation periods of any other rule. Considering that this is indeed a step change in the way safety is being addressed and regulated, “taking our time” is well justified. However, the intention to provide a qualitative assessment of safety (a safety index) might have been enough at the time the probabilistic framework for damage stability was conceived (indeed for as recent as a few years ago) but this is not the case today. With the advent of Design for Safety and of Risk-Based Design, quantification of safety, consistently and accurately, is prerequisite to treating safety as a design objective. This, in turn, entails that the level of detail in the method used to quantify safety carries a much bigger weight. With this in mind and in the knowledge that the Attained Index of Subdivision in the probabilistic rules is a weighted summation of survival factors, calculating survival factors consistently M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_6, © Springer Science+Business Media B.V. 2011

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and accurately is paramount. Unfortunately, a close scrutiny of the work that led to the current formulation of the s-factor revealed that it is simply the result of a series of unjustified compromises, inadvertently creeping in during the rule-making process. In the form, due to be adopted in 17 months time, the s-factor derives from a regression analysis of only a filtered set of old cargo ships and as such it will be unsuitable even for this category of ships, considering the evolutionary changes in most cargo ship types. More importantly, recent research results indicate that, on average, the formulation to be adopted seriously underestimates the inherent survivability of cruise ships whilst it drastically undermines the survivability of Ro-Ro passenger ships. What is of crucial significance, there is little consistency between the actual survivability of both these vessel types (the most safety-critical of ships) and that postulated by the currently proposed formulation. The problem does not end here: since this formulation was used to calculate Index-A for a representative sample of various vessel types so as to evaluate the Required Index of Subdivision (in principle, the safety level to be adopted), these results will also be in disarray. In short, SOLAS 2009 is in need of major introspection! This paper attempts to lay the bare facts and in so doing to raise the alarm, hoping that in the time remaining no effort will be spared in ensuring that the best ever regulatory achievement in our industry will not fall flat on its face before it gets started.

2 The Probabilistic Concept of Ship Subdivision The first probabilistic damage stability rules for passenger vessels, deriving from the work of Kurt Wendel on “Subdivision of Ships”, (Wendel 1968) were introduced in the late sixties as an alternative to the deterministic requirements of SOLAS ‘60. Subsequently and at about the same time as the 1974 SOLAS Convention was held, the International Maritime Organisation (IMO), published Resolution A.265 (VIII). The next major step in the development of damage stability standards came in 1992 with the introduction of SOLAS part B-1 (Chapter II-1), containing a probabilistic standard for cargo vessels, using the same principles embodied in the 1974 regulations. This, in turn, necessitated launching at IMO the regulatory development of “Harmonisation of Damage Stability Provisions in SOLAS, based on the Probabilistic Concept of Survival” in the belief that this represented a more rational approach to addressing damage stability. However, the slowly changing world of IMO engulfed this process and it almost stagnated. In this state of affairs, the compelling need to understand the impact of introduction of probabilistic damage stability regulations in the design of cargo and passenger ships and the growing appreciation of deeply embedded problems in both the rules and the harmonisation process itself, provided the motivation for the adoption of an in-depth evaluation and re-engineering of the whole probabilistic framework through the EC-funded €4.5M, 3-year project HARDER (HARDER

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2003) The overriding goal of this project was to develop a rational procedure for probabilistic damaged ship stability assessment, addressing from first principles all relevant aspects and underlying physical phenomena for all types of ships and damage scenarios. In this respect, HARDER became an IMO vehicle carrying a major load of the rule-development process and fostering international collaboration at its best – a major factor contributing to the eventual success in achieving harmonisation and in proposing a workable framework for damage stability calculations in IMO SLF 47. A stage has now been reached where the draft text of the major revision to the subdivision and damage stability sections of SOLAS Chapter II-1 based on a probabilistic approach has been completed, following final amendments in January 2005 to Regulation 7-1 involving calculation of the “p” factor. The revised regulations were adopted in May 2005 by the IMO MSC and will be entering into force for new vessels with keels laid on or after 1 January 2009. One of the fundamental assumptions of the probabilistic concept of ship subdivision in the proposed regulations is that the ship under consideration is damaged, that is the ship hull is assumed breached and there is (large scale) flooding. This implies that the cause of the breach, the collision event with all the circumstances leading to its occurrence, are disregarded, and hence the interest focuses on the conditional probability of the loss of stability. In other words, risk to life is assumed to be irrelevant on the likelihood of occurrence of a collision event that ends in hull breaching and flooding. For this reason, the regulations imply the same level of “safety” irrespective of the mode of operation that can e.g. take place in area of varying density of shipping (congestion of traffic), or indeed can be so different depending on ship type, or can involve vastly different consequences, etc, all of which might imply considerably different levels of actual risk. This said, all risk-related factors (e.g. size of ship, number of persons on board, life saving appliances arrangement, and so on) are meant to indirectly be accounted for by the Required Index of Subdivision, R. Summarizing, the probability of ship surviving collision damage is given by the Attained Index of Subdivision, A, and is required not to be lesser than the Required Index of Subdivision, R, as given by expression (1): J

I

A   w j  pi  s ji ; A  R

(1)

j 1 i 1

Where: A/R Attained/Required Index of Subdivision j loading condition (draught) under consideration Draught and GM J number of loading conditions considered in the calculation of A (normally 3 draughts) i represents each compartment or group of compartments under consideration for each j

106 D. Vassalos and A. Jasionowski

I wj pi sij

set of all feasible flooding scenarios comprising single compartments or groups of adjacent compartments for each j probability mass function of the loading conditions (draughts) probability mass function of the extent of flooding (that the compartments under consideration are flooded) probability of surviving the flooding of the compartment(s) under consideration, given loading (draught) conditions j occurred

The index A can thus be considered as the expected value of the “s-factor”, with “p- and w-factors” being the respective likelihoods, reflected in worldwide ship collision statistics:

A  E (s )

(2)

Consequently, (1-A) can be considered as the marginal probability of (sinking/ capsize) in these scenarios, and as such it can be used for deriving the collisionrelated risk contribution, (Vassalos 2004). The required Index of subdivision, R (derived principally from regression on A-values of representative samples of existing ships) represents the “level of safety” associated with collision∩flooding events that is deemed to be acceptable by society, in the sense that it is derived using ships that society considers fit for purpose, since they are in daily operation. The new regulations represent a step change away from the current deterministic methods of assessing subdivision and damage stability. Old concepts such as floodable length, criterion numeral, margin line, 1 and 2 compartment standards and the B/5 line will be disappearing. With this in mind, there appears to be a gap in that, whilst development of the probabilistic regulations included extensive calculations on existing ships which had been designed to meet the current SOLAS regulations, little or no effort has been expended into designing new ships from scratch using the proposed regulations. However, attempting to fill this gap through research and though participation in a number of new building projects revealed a more serious problem that affects consistency and validity of the derived results and hence of the whole concept. These findings constitute the kernel of this paper and are described next, following a brief reminder on the derivation in project HARDER and the eventual adoption at IMO of the survival factor “s”.

3 Probability of Survival – “S-Factor” It has to be pointed out that despite the wide-raging investigation undertaken in HARDER, the survival factor “s” adopted in the new harmonised probabilistic rules designates simply the probability of a damaged vessel surviving the dynamic effects of waves once the vessel has reached final equilibrium, post-damage. All other

SOLAS 2009 – Raising the Alarm 107

pertinent factors and modes/stages of flooding are not accounted for. Moreover, whilst considering 7 ship models in the test programme (HARDER 2001), representing a range of different types, sizes and forms, namely: 3 Passenger ships (2 Ro-Ro’s and one cruise liner) and 3 dry cargo ships (Ro-Ro Cargo Ship, Containership, and Bulk Carrier - a Panamax Bulk Carrier was added later), the formulation for the s-factor focused on Ro-Ros and non-Ro-Ros. This categorisation derives primarily from the capsize mechanisms pertinent to these two categories. Therefore, and without labouring the detail of the formulation (for which SSRC was responsible), the following two approaches were followed:

3.1 s-Factor for Low Freeboard Ro-Ro Ships It was always the intention to base the survival factor for Ro-Ro ships on the SEM methodology, (Vassalos et al. 1996). In brief, the method statically develops the volume of water that will reduce the damage GZ curve to exactly zero and at the ensuing critical heel angle, θ, two parameters: h – the dynamic water head and f – the freeboard are calculated as shown in Figure 1. The statistical relationship between parameters (h) and (f) and the significant wave height of the critical survivable sea state (HS) is then established.

GZmax

θ critical

G h f

Fig. 1 The SEM Methodology

108 D. Vassalos and A. Jasionowski

Accounting finally for the likelihood of occurrence of this sea state at the time of casualty led to the following relationship (Figure 2):

s = exp(-exp (f –8h +1.3)

Fig. 2 SEM-Based s-Factor for Ro-Ro Ships (25 Cases – 4 Ship Models)

3.2 s-Factor for non-Ro-Ro Ships Having applied the SEM methodology to non-Ro-Ro ships and achieved what was considered unsatisfactory results, the focus shifted towards the more traditional damage stability criteria employing the use of properties of the GZ curve, such as GZ max, GZ Range, and GZ Area. In simple terms, the correlation of these three traditional parameters with the observed survival sea states from the non-Ro-Ro model tests was first established, shown here in Figures 3 to 5. Range vs. Hs 10 9 8

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SOLAS 2009 – Raising the Alarm 109 Area vs. Hs 10 9 8

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A review of these results led to discarding the GZ area parameter due to lack of correlation as well as most points not falling naturally on the observed trends. Adopting finally an existing SLF format and with Hs limited to 4m, similar to RoRo vessels, yielded the following final formulation (Figure 6):

110 D. Vassalos and A. Jasionowski

s = [ (GZmax / 0.12) (Range / 16) ]1/4

Fig. 6 GZ-Based “s”-Factor for Conventional Ships (25 Cases – 3 Ship Models)

It may be noticed that 25 damage cases were used in the formulations described in the Figure 5 and another 25 damage cases in Figure 6. Thus, the whole data set used in project HARDER, and shown here in Figure 7, comprises 51 data points (the one point not accounted for relates to the cruise ship). 10 Conventional

9

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1

3 2

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1 0

0

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44

0

GZmax [m]

Fig. 7 Data set of 51 measured points of survivability in the HARDER project

Based on the aforementioned developments the following points are noteworthy:  The formulation for the survival factor in the new harmonised probabilistic regulations for damage stability, due to be adopted in January 2009, is based solely on conventional cargo ships, Figure 6. OOOPS!!!  The SEM methodology was never adopted; hence there is no formulation for survival factor in SOLAS 2009 that could be applied to anything other than non-Ro-Ro ships.

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 Examining also Figure 7, leads to some additional and even more disturbing observations:  The single result on the cruise ship, which was never used, shows that even with marginal static stability characteristics the vessel survives sea states with Hs in excess of 4m. Several tests since then clearly demonstrate that the current formulation underestimates the survivability for cruise ships; as a general trend the bigger the vessel the larger the error!  On the other end of the spectrum, the formulation due to be adopted could overestimate the survivability of Ro-Ro ships. In other words, survivability of Ro-Ro ships can be lower than what the formulation implies. Having made these sobering observations, there is yet another blunder to consider: the development of the Required Index, R, for passenger ships. The original results, shown in Figure 8, represent the A-values calculated on the basis of the new formulation for a representative sample of ships. Considering the arguments made above, it may justifiably be proposed that the observed trends simply reflect the gross oversight of the fact that the s-factors considered in the new harmonised rules had nothing to do with passenger ships.

All PASS ships

1,1 1

Rnew

0,9 Rnew (SLF) Rnew (HARDER) Linear (Rnew (SLF)) Linear (Rnew (HARDER))

0,8 0,7 0,6 0,5 0,4

0

25

50

75 100 125 150 175 200 225 250 275 300

Ls (m)

Fig. 8 Rnew for all passenger ships

Not realising this fact, rather than questioning the method of calculation, it was attempted instead to make sense of the results at hand, using an approach questionable at best, which was surprisingly adopted at IMO; the tidal wave of the new probabilistic rules managed to push an excellent framework through the door but it seems that a lot of rumble went through with it.

112 D. Vassalos and A. Jasionowski

4 Emerging Evidence for Raising the Alarm on Solas 2009

4.1 SAFENVSHIP Project In the proposed formulation for the s-factor, described above, the still water GZ curve characteristics are calculated by assuming that the lowest-most spaces within the damaged zone flood instantly and that static balance between vessel mass, displaced water and floodwater describe fully the vessel stability. To gain better understanding of the meaning of the above approximation and, in general, of the stability of modern passenger ships, a study was undertaken, (Jasionowski, 2005), including probabilistic damage stability calculations, numerical time-domain simulations and scaled-model tank testing of three cruise ship designs, considered representative of modern passenger ships. Figure 9 shows a sample result of the study. Altogether 33 different damage cases of the three vessels were investigated, chosen to provide a range of s-factors and distribution of damages along the whole length of the vessel; damage openings were chosen with a level of “high severity” in mind. Deriving from this study the following points are noteworthy:  Of the 33 cases considered, 16 were found to lead to vessel capsizing within two hours, sometimes rapidly. Of the 16 “capsizing” cases, 10 had an s-factor between 0 and 1, with some having s=1.0. Of the remaining 17 “surviving” cases, 7 had an s-factor equal to zero!  The study demonstrates that traditional GZ-curve characteristics cannot adequately describe the behaviour (and hence the destiny) of a damaged ship with the complexity in watertight subdivision of internal spaces found on a typical cruise vessel, and consequently, that the formulation for the s-factor in its present form cannot meaningfully represent the average resistance of such ships to capsize when subjected to flooding, following collision damage. Another reason for this deficiency derives from explicitly neglecting the presence of multiple free surfaces (MFS? effect), a phenomenon specific to ships with complex watertight subdivision, which substantially weakens or indeed leads to complete erosion of stability at any of the stages of flooding (this is a finding postHARDER).

SOLAS 2009 – Raising the Alarm 113 C1 - STD Opened 1 0.9 0.8 0.7 0.6 0.5

A=0.638

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Fig. 9 A map of “s” factors (upper plot) and the expression p.(1-s) for all damage cases considered as a function of the damage locations. The red boxes in the lowest-most plot indicate cases where the ship capsized within 2 hours simulation time, whereas green boxes indicate survival cases

4.2 Newbuilding Design Experience With the introduction of the new probabilistic rules, the dilemma of prescriptive SOLAS-minded designers must be overcome. Using an approach, known as platform optimisation, SSRC/SaS developed a fully automated optimisation process leading to ship designs with an acceptable level of flooding-related “risk” (i.e., compliance with statutory requirements) that can be optimised along with cost-effectiveness, functionality and performance objectives. Alternatively, high survivability internal ship layouts can be developed, without deviating much from the current SOLAS practice, thus making it easier for ship designers to relate to the proposed procedure. This procedure has been extended to address in full the damage survivability of new building cruise and RoPax vessels from first principles in a systematic and all embracing way, as explained in (Vassalos 2007). Moreover, a direct link between Index-A and flooding risk has been established, thus facilitating a better understanding of the meaning of all the factors involved and of their significance in affecting safety. This also led to new evidence concerning the inconsistency and inaccuracy in the details of the new probabilistic rules, as summarised in Figure 10.

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Fig. 10 Cumulative probability distributions for time to capsize for a traditional Ro-Ro vessel and a large new building cruise ship. Deepest draft loading conditions.

The cumulative distributions for time to capsize, for the ships shown in Figure 10, are derived using two approaches: (a) an analytical expression based on the formulation of the new probabilistic rules and (b) time-domain numerical simulations of 500 damage cases derived my means of Monte Carlo sampling that represent the statistics of damage characteristics used in these rules. Both ships comply with the new rules. Moreover, as explained in (Vassalos 2007), the values of these distributions tend to 1-A within a reasonable time period of several hours. Therefore, based on these results which seem to be typical for these groups of vessels, the following observations can be made:  For the Ro-Ro vessel, (Jasionowski 2007), approximately one in three collision ∩flooding events would lead to capsize within an hour as inferred from either analytical solution or numerical simulation (this is not a good standard!)  Application of the new rules indicates that the cruise ship will follow a similar fate in approximately one every 10 events whilst using first-principles timedomain simulation tools the rate becomes approximately 1:100. This is a big difference! These and other similar results have alerted the industry to the extent that a new investigation is being launched in an attempt to sort out the details in the new rules before they are adopted in January 2009, in the belief that the probabilistic rules constitute one of the most significant developments in the rule-making history of the maritime profession.

5 Conclusions Based on the results and arguments presented in this paper the following conclusions can be drawn:

SOLAS 2009 – Raising the Alarm 115

 The index A of new probabilistic rules on ship subdivision can be interpreted as a value that reflects the average survivability of a vessel following collision damage and flooding in a seaway. As such, an accurate calculation of the survival probability in these rules is of paramount importance.  This being the case, there is new evidence mounting that indicates gross errors in the derivation of survival factors, demanding swift action by the profession to avert “embarrassment” in global scale.  Such action is already being planned but it needs a wider participation to ensure that results will be available on time to re-engineer a correct formulation before the new rules are enforced.  More importantly, in the knowledge that the probabilistic rules are inconsistent and inaccurate, due care is needed in designing new ships; in particular, the use of time-domain simulation tools or physical model tests must be fully exploited.

Acknowledgments The financial support of the UK Maritime and Coastguard Agency and of the European Commission in the research undertaken is gratefully acknowledged. Special thanks are due to Royal Caribbean International and to Carnival for providing the inspiration needed to delve beyond trodden paths, even when it is not called for.

References HARDER (2001) Task 3.1 – Main Report of Model Tests. Report Number 3-31-D-2001-13-1 HARDER (2003) Harmonisation of Rules and Design Rationale. U Contact No. GDRB-CT1998-00028. Final Technical Report Jasionowski A, (2005) Survival Criteria For Large Passenger Ships. SAFENVSHIP Project, Final Report, Safety at Sea Ltd Jasionowski A, York A, (2005) Investigation Into The Safety Of Ro-Ro Passenger Ships Fitted With Long Lower Hold. UK MCA RP564, Draft Report MCRP04-RE-001-AJ, Safety at Sea Ltd Vassalos D (2004) A risk-based approach to probabilistic damage stability. 7th IntWorkshop on the Stab and Oper Safety of Ships, Shanghai, China Vassalos D (2007) Risk-Based Design: Passenger Ships. SAFEDOR Mid-Term Conf, the Royal Inst of Naval Archit, Brussels Vassalos D, Pawlowski M, Turan O (1996) A Theoretical Investigation on the Capsizal Resistance of Passenger/Ro-Ro Vessels and Proposal of Survival Criteria. Joint Northwest European Project Wendel K (1968) Subdivision of Ships. Diamond Jubilee Int Meeting, N.Y.

2 Stability of the Intact Ship

Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas A. Yucel Odabasi and Erdem Uçer Istanbul Technical University, Faculty of Naval Architecture and Ocean Engineering

Abstract This study intended to evaluate the behavior of a biased ship under main, sub and super harmonic resonant excitation. For the sake of simplicity Duffing’s equation is used for un-biased roll equation. In the first part of this study, first and second order approximate solutions of biased roll equation have been obtained for different equation parameters in main, sub and super harmonic resonance regions by using method of multiple scales and Bogoliubov-Mitropolsky asymptotic method. It was found that second order approximate solutions had a better compliance with numerical results when the solution is stable. In the second part, stable solution bounds of the biased roll equation were obtained for different equation parameters by using numerical methods. It was found that the size of the bounds are highly dependent on initial bias angle, linear damping coefficient, amplitude and frequency of wave excitation and phase angle of the excitation force and also the symmetry of the bounds were dependent on the magnitude of the initial bias angle. From the results obtained it can be concluded that Lyapunov’s stability theory provides the most reliable results.

1 Introduction Resonance which is an important phenomenon can lead to capsize of ships because it causes amplitude of rolling motion to take large values after a few cycles. Thus, many scientists and researchers investigated resonance case in the rolling motion by either making scaled model test or using asymptotic, perturbation and numerical methods and also Melnikov Method. Wright and Marshfield (1980) presented experimental and theoretical study on ship roll response and capsize behaviour in beam seas. They made experiments to measure the steady state sinusoidal roll amplitudes for three type of models (Low, Medium and High freeboard model) at a range of frequencies with constant wave slope amplitude inputs. In their theoretical study, they used regular perturbation M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_7, © Springer Science+Business Media B.V. 2011

119

120 A.Y. Odabasi and E. Uçer

method, harmonic balance method and the method of averaging to solve roll equation with or without bias. Another important experimental study was made by Grochowalski (1989). He presented experimental study on the mechanism of ship capsizing in quartering and beam waves. He used a 1:14 scaled GRP model which was fitted with bulwarks, freeing ports, superstructure and stern ramp and also had a large centreline skeg and a single four blade propeller. He made experiments for two load conditions (light and full load condition). He used free model runs to give an insight into the kinematics of ship rolling and also captive tests to identify the composition of wave exciting hydrodynamic forces and moments. In his study, he indicated the danger created by bulwark submergence and accumulation of water on deck and also stability reduction on a wave crest. In his experiments he also found that the quartering waves could be as dangerous as beam waves. Cardo, Francescutto and Nabergoj (1981) used Bogoliubov-Krylov-Mitropolsky asymptotic method to determine first order steady state solution of roll equation without bias in ultra harmonic and sub harmonic resonance regions. (Nayfeh and Khdeir 1986) used method of multiple scales to determine a second order approximate solution in the main resonance region for the nonlinear harmonic response of biased ships in regular beam waves. Nayfeh (1979) obtained the first and second order solutions of Equation (1) in main, sub harmonic and super harmonic resonance regions with the method of multiple scales. This equation is similar to the equation used in this article. However, the solutions obtained in this article are different from Nayfeh’s solutions due to transformations made in Equation 1.

~ x  2 ˆ x  02 x   2 x 2   3 x 3  K Cos  t

(1)

Jiang et al. (1996) investigated the influence of bias on capsizing both in terms of the reduction of the safe basin and in terms of phase space flux. Their conclusions were based on Melnikov analyses and also supported by extensive numerical solution. Macmaster and Thompson (1994) examined capsize of ships excited by the naturally propagating wavefront created by a laboratory wave maker that is suddenly switched on. They used transient capsize diagrams and showed the importance of bias. Cotton et al. (2000) used the same equation shown below to examine the sensitivity of capsize to a symmetry breaking bias in beam seas and stated that the case α=0 to be a sensible model of ship roll motion.

x   x  x (1  x ) (1   x )  F Sin  t

(2)

Spyrou et al. (1996) used Melnikov method to obtain analytical formulas which can characterize capsizability of ships taking into account the presence of some roll bias. They have tested these formulas against numerical simulations targeting

Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 121

the process of safe basin erosion. Their mathematical model for rolling motion was also Equation (2). The aim in the first part of this study is to present available methods in the solution of roll equation (method of multiple scales, Bogoliubov-Mitropolsky asymptotic method and numerical methods), compare the obtained results and also show the sensitivity of solution to the effect of equation parameters. It was also observed from the solutions that neglecting the effect of cubic coefficient of restoring term is not sensible. In the second part of this study, stable solution bounds of the biased roll equation were obtained for different equation parameters and the effect of these parameters on the size and symmetry of the bounds were also examined. In the concluding part of the paper it is stated that boundedness of solution, in the sense of Lyapunov, constitute a sound basis for the intact stability research.

2 Methodology of the Investigation

2.1 Equation of Motion and Preliminary Considerations In this paper, while the rolling motion of the ship was being modelled, interactions between rolling and other modes of motion have been ignored and the ship was considered to have a rigid body and sea water was ideal and incompressible. Under these assumptions, rolling motion of a ship was written as in Equation (3).

 

I   D  ,   M R    E t   M wind    

(3)

θ : Rolling angle with respect to calm sea surface  : Roll angular velocity I : Mass moment of inertia including the added mass moment of inertia. MR(θ) : Restoring moment. Its coefficients can be obtained either from a computed righting arm curve by curve fitting or from the characteristics of the righting arm curve, such as GM, GZmax, area under the righting arm curve, by constrained curve fitting.

 D(θ,  ) : Damping moment. In this study, the linear damping moment associated with wave produced by the hull was only taken care of.  E(t) : Roll exciting moment, due to external force. When the wave elevation is induced by regular transversal waves, the exciting term may be written as E t   Ew Cos t  where Ew is the amplitude and Ω the angular frequency Mwind : Wind Moment

122 A.Y. Odabasi and E. Uçer

In this study, for the sake of simplicity, the following approximations have been used M R ( )   k1  k 2 3 and D( , )  s  where Δ is the displacement and s is the linear damping coefficient Under the above assumptions Equation (3) can be written as









I   s    k1  k 2 3  Ew Cos  t  M wind

(4)

Dividing both sides of the equation by the mass moment of inertia, the expression given in Equation (5) is obtained.

  ˆ   c1  c2 3  F Cos t  Fruz

(5)

This type of roll equation was also examined from other author’s earlier studies. If θ = x + θs transformation is made, Equation (5) can be re-written as

x  ˆ x  02 x  d1 x 2  d 2 x 3  F Cos t 2

2

(6) 3

where 0  c1  3 c 2  s , d1  3  s c 2 , d 2  c 2 , Fruz  c1 s  c 2  s In the following parts of this paper Equation (6) were used as simplified rolling equation.

2.2 Main, Sub and Super Harmonic Resonance Investigation In this study, wave excitation was assumed to be approximately half, twice and equal to the natural frequency to make an investigation in sub, super and main harmonic resonance regions. Then biased roll equation was solved in these regions by using the method of multiple scales, Bogoliubov-Mitropolsky asymptotic method and numerical methods. The effects of parametric resonance have not been investigated. Finally, the stable solution bounds of roll motion were obtained for different bias angles, linear damping coefficients, and amplitudes and frequencies of wave excitation in the sub, super and main harmonic resonance regions by using Mathematica NDSolve routine and the Fortran program written by the authors. The necessary data used in numerical computation were shown in Table 1.

Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 123 Table 1 The data used in numerical calculations. θs (rad.) ω0

d1

ˆ

d2

F

0.005

0.610257 0.001975

0.100

0.607020 0.039499 0.131664 0.04,0.07,0.10,0.13 0.08,0.10,0.12

0.200

0.597179 0.078998

2.3 Solution Methods 2.3.1 Method of Multiple Scales Method of multiple scales assumes that the expansion representing the response to be a function of multiple independent variables, or scales instead of a single variable. In this process, first the new independent variables are introduced.

Tn   n t For n=0,1,2…… where ε is a small parameter Time derivatives then can be re-written as

d d T0  dT    1  ........  D0   D1  ........ dt d t  T0 d t  T1

(7a)



(7b)



d2  D02  2  D0 D1   2 D12  2 D0 D2  .......... d t2

To illustrate the implementation of the method first order approximate solution of Equation (6) in the main resonance region is provided below. In the main resonance region where the excitation frequency is near the natural frequency of the ship (    0    ), the linear damping coefficient, quadratic and cubic restoring moment coefficients and finally amplitude of the excitation frequency were considered to have order of є. Thus, Equation (6) was written in as;





x  02 x    x  d x 2  g x 3 

 f 2

e 

i  0T0  T1 

 e  i  0T0  T1 



(8)

If the expressions shown in Equation (7a) and (7b) are substituted into Equation (8), Equation (9) is obtained.

124 A.Y. Odabasi and E. Uçer

D02 x  2  D0 D1 x   2 D12 x  2 2 D0 D2 x  02 x    D0 x   2  D1 x   d x 2   g x3 

 f 2

e 

i  0 T0    T1 

(9)



 e  i  0T0    T1   ......

The solution of above equation can be represented by an expansion as

x  x0 (T0 , T1 , T2 ,....)   x1 (T0 , T1 , T2 ,....)

(10)

  2 x2 (T0 , T1 , T2 ,....)  ...........

If the expansion shown in Equation (10) is substituted into Equation (9) and coefficients of like powers of ε are equated, we obtain the following equations.

 0 : D02 x0  02 x0  0

(11a)

 1 : D02 x1  02 x1  2 D0 D1 x0   D0 x0  d x02  g x03 





(11b)

 D12 x0   D0 x1   D1 x0

(11c)

f i0T0 iT1 e e  e i0T0 e iT1 2

 2 : D02 x2  02 x2  2 D0 D1 x1  2 D0 D2 x0  2d

x0 x1  3 gx02

x1

The solution of Equation (11a) is

x0  A(T1 ) ei 0T0  A (T1 ) e i  0T0 where A(T1 ) 

(12)

1 1 a(T1 ) ei  (T1 ) and A (T1 )  a(T1 ) e  i  (T1 ) , A and A are the 2 2

only functions of T1 as we are searching for first approximate solution. If x0 solution shown in Equation (12) is substituted into Equation (11b), a new differential equation is obtained. The right hand side of this equation includes secular terms. These secular terms are eliminated by equating the coefficients of e i 0 T0 and e  i 0 T0 to zero.

Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 125

f iT1 e 0 2

(13a)

f iT1 e 0 2

(13b)

 2 i 0 D1 A  i  0 A  3 A A2 g  2 i 0 D1 A   i  0 A  3 A A 2 g 

If A and A are substituted into the above equations, after some transformations which will not be shown here due to limited space, Equation (14a) and (14b) are obtained.

a 

a 2

a  



f 2 0

Sin T1     0

3g 3 f a  Cos T1     0 8 0 2 0

(14a)

(14b)

where (΄) symbol represents derivative taken with respect to T1. Defining  T1     and        , the above equation becomes

a  

a 2



f 2 0

Sin  

(15)

The above equations can also be written in the form of the equations below.

 a  f  a    Sin   2 0  2 

(16a)

 3g 3  f a    a     a  Cos   2 0  8 0 

(16b)

The first order approximate solution of biased roll equation is written as:

x  a Cos   t       

(17)

where a and γ can be found from Equation (16a) and (16b). This method had been used to obtain an approximate solution of roll equation from many scientist and researchers (Nayfeh 1979, Nayfeh and Khdeir 1986).

126 A.Y. Odabasi and E. Uçer

Thus, second order approximate solution and also solution in sub and super harmonic resonance regions weren’t explained here. Due to decrement in linear damping coefficient or increment in initial bias angle and amplitude of wave excitation, the magnitude of coefficient “a” in the steady state solution increases in main, sub and super harmonic resonance regions. As can be seen from Fig. 1 and 2, when the wave excitation frequency is approximately half of the natural frequency, coefficient “a” takes its value in negative small values of σ (deviation between excitation frequency and natural frequency). The jump phenomenon was also observed when linear damping coefficient µ was equal to 0.04 and amplitude of the wave excitation (F) taken value higher or equal to 0.12. m=0.04 F=0.12 d2=0.131664 0.6 0.5

Bias=0.20 rad. Bias=0.15 rad. Bias=0.10 rad.

a

0.4 0.3 0.2 0.1 0 -0.15 -0.12 -0.09 -0.06 -0.03

0 s

0.03 0.06 0.09 0.12 0.15

Fig. 1 a-σ graph The effect of initial bias angle w0=0.597179 d1=0.078998 d2=0.131664 m=0.04 0.6

F=0.12 F=0.1 F=0.08

0.5

a

0.4 0.3 0.2 0.1 0 -0.15 -0.12 -0.09 -0.06 -0.03

0 s

0.03 0.06 0.09 0.12 0.15

Fig. 2 a-σ graph The effect of amplitude of excitation

As can be seen from Fig. 3, if the wave excitation frequency is approximately twice of the natural frequency, coefficient a in the first and second approximate solution of rolling equation goes to directly to zero without any oscillation when time goes to infinity. If the linear damping coefficient is increased, coefficient “a” goes to zero faster.

Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 127

Fig. 3 a-t graph (µ=0.04, F=0.12, σ =-0.05,

  2  0 , θs=0.2 rad.)

As can be seen from the Fig. 4 and Fig. 5, while the other parameters of the equation are fixed if we take σ =-0.1 instead of σ =0.1 in the main resonance region, the coefficient “a” gets much larger values (nearly twice of the former).

Fig. 4 a-t graph (µ=0.04, F=0.12, σ =0.1,

   0 , θs=0.2 rad.)

Fig. 5 a-t graph (µ=0.04, F=0.12, σ =-0.1,

   0 , θs=0.2 rad.)

2.3.2 Bogoliubov-Mitropolsky Asymptotic Method While examining the main resonance region by Bogoliubov-Mitropolsky asymptotic method, ω02=Ω2+εΔ, linear damping coefficient, quadratic and cubic term of restoring

128 A.Y. Odabasi and E. Uçer

moment and finally amplitude of excitation force have been assumed to be order of ε. Under these assumptions biased roll equation were written as



x   2 x     x  d x 2  g x 3  f Cos t   x



(18)

The above equations can be written in more efficient form as

   x  d x 2  g x 3   x   2 x      f Cos Cos  f Sin Sin   x   

(19)

where ψ = Ω t+β The solution is assumed to have the form (Bogoliubov and Mitropolsky 1961),

x  a Cos   u1  a,  t ,   ...........

(20)

and a and β in the above equation must satisfy the following expressions

da   A1 (a,  )   2 A2 (a,  )  ......... dt

(21a)

d   B1 (a,  )   2 B2 (a,  )  .......... dt

(21b)

To obtain first order approximate solution of biased roll equation, only first terms were taken in the right hand side of the Equation (20), (21a) and (21b). If the first term of expression shown in Equation (20) is put into the left hand side of the Equation (19), Equation (22) is obtained. If it is put into right hand side of the Equation (19), Equation (23) is obtained.

x   2 x    2 A1 Sin  2 a  B1Cos   a cos





   x  d x 2  g x 3  f CosCos  f Sin Sin   x  d a2   a Cos   a  Sin  2   2 3 d a g a   2 Cos 2  4 Cos 3 

 3 g a3  f CosCos  f Sin Sin  4 

(22)

 (23)

Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 129

If the right hand sides of the Equation (22) to Equation (23) are equated and the terms which have Cosψ and Sinψ are brought together, the following expressions are obtained, and with the help of other terms, u1 solution can be found as:

 2  A1   a   f Sin 

 2 a  B1    a 

u1 

3 g a3  f Cos  4

(24a)

(24b)

d a2 d a2 g a3  Cos 2   Cos3 2 2 6 2 32  2

(25)

If A1 and B1 are substituted into the Equation (21a) and Equation (21b), the following expressions are obtained:

da ˆ a F   Sin  dt 2 2 d  02   2 3 d 2 a 2 F    Cos  dt 2 8 2a

(26a)

(26b)

The first order approximate solution of biased roll equation is the given as:

x  a Cos  t       

(27)

where a and β coefficients are determined from Equation (26a) and (26b). In sub and super harmonic resonance regions instead of using Equation (18), F Equation (28) is used. Equation (28) is obtained by making x  y  2 Cos  t 0   2 transformation in the Equation (6).

  y     Sint  d  y   Cos  t 2  y  02 y    0 3  g  y   Cos  t   where  

F   2 2 0

(28)

130 A.Y. Odabasi and E. Uçer

2.3.3 Numerical Methods Biased roll equation was solved by Mathematica 5-NDSolve with 50000 steps in sub, super and main harmonic resonance regions for different initial conditions and parameters of the equation. The results also controlled with Fortran IVPAG routine. Stable bounds for different equation parameters were obtained by Mathematica 5-NDSolve and the program (“Rk.f90”) written in Fortran 90 language by the authors. “Rk.f90” uses Runge Kutta Fourth order method with time step 0.001 second to solve biased roll equation and finds out the initial conditions cause to solution goes to infinity when time goes to infinity. Solutions of “Rk.f90” program are the same as the solutions obtained by Mathematica 5-NDSolve.

3 Compliance of Perturbation & Asymptotic Results with Numerical Solution The solutions obtained by using method of multiple scales (MSM) shown in Fig. 6-11 in and Bogoliubov-Mitropolsky asymptotic method solutions shown in Fig. 12-17. As can be seen from the figures, the second order approximate solutions obtained by using MSM and asymptotic method in sub, super and main harmonic resonance regions, have better compliance with numerical solution rather than the first order approximate solutions. Second order approximate solutions only have difference with numerical solutions in transient region. However, the first order approximate solutions also have differences in steady state region. Although this good compliance of perturbation and asymptotic results with numerical solution in stable bounds, perturbation and asymptotic methods could not explain unstable cases. 1.5

ω=0.597179 d=0.078998 σ=0.1 F=0.12

MSM O(ε^2) Numeric

1

x(t)

0.5 0

0

50

100

150

200

250

300

350

400

-0.5 -1 -1.5 t

Fig. 6 Comparison of second order approximate MSM and numeric solution when    0

Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 131 1.50

x(0)=0 dx/dt=0.56 ω=0.59718 Ω=1.29436 d=0.078998

MSM O(ε^2) Numeric

1.00

x(t)

0.50

0.00 50

0

100

150

200

250

300

350

400

-0.50

-1.00

Fig. 7 Comparison of second order approximate MSM and numeric solution when   20 1

MSM O(ε^2) Numeric

x(0)=0.08045 dx/dt=0.21482 ω=0.59718 Ω=0.28359 d=0.078998

0.8 0.6 0.4

x(t)

0.2 0

0

50

100

150

200

250

300

350

400

-0.2 -0.4 -0.6 -0.8 -1 -1.2 t

Fig. 8 Comparison of second order approximate MSM and numeric solution when    0 2 1.5

x(0)=0 dx/dt=0.459 ω=0.59718 Ω=0.69718 d=0.078998

MSM O(ε) Numeric

1

x(t)

0.5

0

0

50

100

150

200

250

300

350

400

-0.5

-1

-1.5 t

Fig. 9 Comparison of first order approximate MSM and numeric solution when    0

132 A.Y. Odabasi and E. Uçer 1.5

x(0)=0 dx/dt=0.56 ω=0.59718 Ω=1.29436 d=0.078998

MSM O(ε) Numeric

1

x(t)

0.5

0

0

50

100

150

200

250

300

350

400

-0.5

-1

t

Fig. 10 Comparison of first order approximate MSM and numeric solution when   2 0 1

x(0)=0.1 dx/dt=0.2 ω=0.59718 Ω=0.2835 d=0.078998

0.8

MSM O(ε) Numeric

0.6 0.4

x(t)

0.2 0 -0.2

0

50

100

150

200

250

300

350

400

-0.4 -0.6 -0.8 -1 -1.2

t

Fig. 11 Comparison of first order approximate MSM and numeric solution when    0 2 1.5

μ=0.1,F=0.12, Ω=0.697179, ω=0.597179 x(0)=0 dx/dt=0.535

Numeric Asymptotic O(ε^2)

1

x(t)

0.5

0

0

50

100

150

200

250

300

350

400

-0.5

-1 t

Fig. 12 Comparison of second order approximate asymptotic and numeric solution when    0

Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 133 1.5

μ=0.04,F=0.12, Ω=1.294358,ω=0.597179 x(0)=0 dx/dt=0.56 Bias=0.2 rad.

Numeric Asymptotic O(ε^2)

1

x(t)

0.5

0

0

50

100

150

200

250

300

400

350

-0.5

-1 t

Fig. 13 Comparison of second order approximate asymptotic and numeric solution when   20 1.5

Ω=0.285755 F=0.12 x(0)=0.25 y(0)=0.268 Bias=0.2 rad.

Asymptotic O(ε^2) Numeric

1

x(t)

0.5 0 0

50

100

150

200

250

300

350

400

-0.5 -1 -1.5 t

Fig. 14 Comparison of second order approximate asymptotic and numeric solution when 1.5

  0 2

μ=0.1,F=0.12, Ω=0.697179,ω=0.597179 x(0)=0 dx/dt=0.535

Asymptotic O(ε) Numeric

1

x(t)

0.5 0

0

50

100

150

200

250

300

350

400

-0.5 -1 -1.5 t

Fig. 15 Comparison of first order approximate asymptotic and numeric solution when    0

134 A.Y. Odabasi and E. Uçer 1.5

x(0)=0 y(0)=0.56 ω=0.597179 Ω=1.29436 Bias=0.2 rad.

Asymptotic O(ε) Numeric

1

x(t)

0.5

0

0

100

50

150

200

250

300

350

400

-0.5

-1 t

x(t)

Fig. 16 Comparison of first order approximate asymptotic and numeric solution when   20 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 0 -0.4 -0.6 -0.8 -1 -1.2 -1.4

ω=0.597179 Ω=0.285755 Bias=0.2 rad.

50

100

150

200

250

300

Asymptotic O(ε) Numeric

350

400

t

Fig. 17 Comparison of first order approximate asymptotic and numeric solution when    0 2

4 Existence of Bounds for Initial Conditions and its Dependence to Parameters As can be seen from Table 2, in the main resonance region, when σ = -0.1 (σ: Deviation of excitation frequency from natural frequency) and |F| ≥ 0.1 (F: Amplitude of wave excitation), there is no stable bound. If amplitude of wave excitation is decreased to 0.08, there are stable bounds for very large linear damping coefficients. The effect of sign and magnitude of σ can easily be seen from Table 3 When σ = 0.1, there are stable bounds even for |F|=0.12. The increment of the amplitude and initial bias angle or decrement of linear damping coefficient causes the narrowing of stable solution bounds.

Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 135 Table 2 Stable bounds in main resonance region for σ = -0.1 Amplitude of

Initial Bias Angle

Initial Bias Angle

Wave excitation

θs=0.2 rad.

θs=0.1 rad.

Initial Bias Angle θs=0.005 rad.

|F|=0.12

-

-

-

|F|=0.10

-

-

-

|F|=0.08

-

Stable Bound

Stable Bound

for µ ≥0.13

for µ ≥0.1

Table 3 Stable bounds in main resonance region for σ = 0.1 Amplitude of

Initial Bias Angle Initial Bias Angle Initial Bias Angle

Wave excitation θs=0.2 rad. F=±0.12 F=±0.10 F=±0.08

θs=0.1 rad.

θs=0.005 rad.

Stable Bound

Stable Bound

Stable Bound

For µ > 0.10

For µ ≥ 0.07

For µ ≥ 0.04

Stable Bound

Stable Bound

Stable Bound

For µ > 0.07

For µ > 0.04

For µ ≥ 0.04

Stable Bound

Stable Bound

Stable Bound

For µ > 0.04

for µ ≥0.04

for µ ≥0.04

There is another factor which effects size of stable bounds. That is the phase angle which will be given to excitation force. The stable bound obtained for μ=0.1, d1=0.039499, d2=0.131664, ω0=0.60702, F=±0.12 and σ = 0.1 shown in Table 4. Table 4 Boundaries of stable bound for μ=0.1 |F|=0.12 θs=0.1 rad. σ =0.1 F=0.12

µ=0.1

F=-0.12

µ=0.1

x (rad.)

x (rad/s)

x (rad.)

x (rad/s)

0.000

0.634

0.000

0.844

0.200

0.605

-0.200

0.854

0.400

0.558

-0.400

0.842

0.600

0.491

-0.600

0.806

0.800

0.404

-0.800

0.742

1.000

0.298

-1.000

0.636

1.200

0.170

-1.065

0.587

1.427

0.000

-1.065

0.000

Let’s take a point which is inside the stable bound shown in Table 4. For example,

x (0)  0 and x (0)  0.634 . If this point is taken as the initial condition, some points on the solution orbit in x- x phase space are out of the stable bound (see Table 5 and Fig. 18).

136 A.Y. Odabasi and E. Uçer Table 5 Some points out of the stable bound tt5.24 t (sn.)

x (rad.)

x

19.2

-0.033582

0.764827

19.3

0.042842

0.763051

19.4

0.118912

0.757785

19.5

0.194286

0.749140

19.6

0.268632

0.737270

(rad/s)

Fig. 18 Solution orbit in x- x phase space for initial condition x(0)=0 x (0)=0.634

As it can be seen from Fig. 19, if x(0)=0.194286 x (0)=0.749140 are taken as initial condition and solve the biased roll equation, the solution is unstable (when (when time goes to infinity, solution also goes to infinity). y 1.8 1.6 1.4 1.2 0.8

1

2

3

4

x

0.6

Fig. 19 Solution orbit in x- x phase space for initial condition x(0)=0.194286 x (0)=0.749140

The solution is stable for x(19.5)=0.194286 and y(19.5)=0.74914 initial condition, Fig. 20.

Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 137

 phase space for initial condition x(19.5)=0.19428 Fig. 20 Solution orbit in x- x

x (19.5)=0.74914

The solution was showed unstable for x(0)=0.194286 x (0)=0.74914 initial condition. If δ=1.220519 rad. phase angle is put into the excitation force and the biased roll equation is solved for x(0)=0.194286 x (0)=0.749140 initial condition again, the solution is stable (see Fig. 21). Hence, the effect of phase angle which will be given the excitation force were proved.

Fig. 21 Solution orbit in x- x phase space for initial condition x(0)=0.194286

x (0)=0.74914 δ=1.220519 rad. In sub and super harmonic resonance regions, the increment of the amplitude and initial bias angle or decrement of linear damping coefficient causes to narrow of stable solution bounds and also the phase angle which will be given to excitation force effects size of stable bounds like main resonance region. Some of the stable bounds obtained when the excitation frequency is approximately half of the natural frequency are shown in Fig. 22-24. As can be seen from these figures, the symmetry of the stable bounds corrupts after the initial bias angle increases. Fig. 25 shows sample of orbit for frequency equal to half the natural frequency.

138 A.Y. Odabasi and E. Uçer

Fig. 22 Stability bound for µ=0.04 θs=0.20 rad. |F|=0.12 σ =-0.078

Fig. 23 Stability bound for µ=0.04 θs=0.10 rad. |F|=0.12 σ =-0.04

Fig. 24 Stability bound for µ=0.04 θs=0.005 rad. |F|=0.12 σ =-0.03

Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 139

Fig. 25 Sample solution orbit in x- x phase space for    0 2

5 Concluding Remarks Due to decrement in linear damping coefficient or increment in initial bias angle and amplitude of wave excitation, the magnitude of coefficient a in the steady state solution increases in main, sub and super harmonic resonance regions. When the wave excitation frequency is approximately half of the natural frequency, coefficient “a” takes its maximum value in negative small values of σ (Deviation between excitation and natural frequency) and also there is jump phenomenon when linear damping coefficient is equal to 0.04 and amplitude of the excitation is higher or equal to 0.12. If the wave excitation frequency is approximately twice of the natural frequency, coefficient “a” in the first and second approximate solution of rolling equation goes to directly to zero without any oscillation when time goes to infinity. If the linear damping coefficient, is increased coefficient “a” goes to zero faster. While the other parameters of the equation are fixed if σ =-0.1 is taken instead of σ =0.1 in the main resonance region, the coefficient “a” gets much bigger values (nearly twice of the former). The second order approximate solutions of asymptotic method and the method of multiple scales have better compliance with numerical solutions in main, sub and super harmonic resonance rather than first order approximate solution of the asymptotic method and the method of multiple scales. The second order approximate solutions only have differences with numerical solutions in transient part of the solutions. However, the first approximate solutions also have differences with numerical solutions in steady state part of the solutions. Although this good compliance of perturbation and asymptotic results with numerical solution in stable bounds, perturbation and asymptotic methods could not explain unstable cases well.

140 A.Y. Odabasi and E. Uçer

The magnitude of linear damping coefficient, initial bias angle, amplitude and frequency of wave excitation and the phase angle which are given to excitation force effects the size of stable solution bounds in main, sub and super harmonic resonance regions. The increment of the amplitude and initial bias angle or decrement of linear damping coefficient causes the narrowing of stable solution bounds. The phase angle of the excitation force can make an unstable initial condition stable initial condition. Thus, importance of this parameter shouldn’t be neglected. The initial bias angle is also affected on the symmetry of the stable solution bounds. The increment of initial bias angle causes to increase asymmetries.

References Bogoliubov N N, Mitropolsky Y A (1961) Asymptotic methods in the theory of nonlinear oscillations. Intersciense NY Cardo A, Francescutto A et al. (1981) Ultra harmonics and subharmonics in the rolling motion of a ship: steady state solution. Int Shipbuild Prog 28:234-251 Cotton B, Bishop S R et al. (2000) Sensitivity of capsize to a symmetry breaking bias. In: Vassalos D, Hamamato M et al. (eds). Contemp Ideas on Ship Stab. Elsevier Oxford Grochowalski S (1989) Investigation into the physics of ship capsizing by combined captive and free running model tests. Trans of Soc of Nav Archit and Mar Eng 97:169-212 Jiang C, Troesch A W et al. (1996) Highly nonlinear rolling motion of biased ships in random beam seas. J of Ship Res 40 2:125-135 Macmaster A G, Thompson J M T (1994) Wave tank testing and the capsizability of hulls. Proc of the R Soc London A 446:217-232 Nayfeh A H (1979) Nonlinear oscillations. John Wiley NY Nayfeh A H, Khdeir A A (1986) Nonlinear rolling biased ships in regular beam waves. Int Shipbuild Prog 33:84-93 Spyrou K J, Cotton B et al. (1996) Analytical expressions of capsize boundary for a ship with roll bias in beam waves. J of Ship Res 46 3:125-135 Wright J H G, Marshfield W B (1980) Ship roll response and capsize behavior in beam seas. Trans of RINA 122:129-149

Historical Roots of the Theory of Hydrostatic Stability of Ships Horst Nowacki* Larrie D. Ferreiro** * Technical University of Berlin (Germany), ** Defense Acquisition University and Catholic University of America (United States)

Abstract The physical principles of hydrostatic stability for floating systems were first pronounced by ARCHIMEDES in antiquity, although his demonstration examples were limited to simple geometrical shapes. The assessment of stability properties of a ship of arbitrary shape at the design stage became practically feasible only about two millennia later after the advent of infinitesimal calculus and analysis. The modern theory of hydrostatic stability of ships was founded independently and almost simultaneously by Pierre BOUGUER (“Traité du Navire”, 1746) and Leonhard EULER (“Scientia Navalis”, 1749). They established initial hydrostatic stability criteria, BOUGUER’s well-known metacenter and EULER’s restoring moment for small angles of heel, and defined practical procedures for evaluating these criteria. Both dealt also with other aspects of stability theory. This paper will describe and reappraise the concepts and ideas leading to these historical landmarks, compare the approaches and discuss the earliest efforts leading to the practical acceptance of stability analysis in ship design and shipbuilding.

1 Introduction Human awareness of the significance of ship stability for the safety of ocean voyages is probably as ancient as seafaring. An intuitive, qualitative understanding of stability and of the risks of insufficient stability must have existed for millennia. The foundations for a scientific physical explanation and for a quantitative assessment of hydrostatic stability were laid by ARCHIMEDES in antiquity (Archimedes 2002, “On Floating Bodies”). Yet despite many important contributions and partially successful attempts by scientists in the early modern era like STEVIN, HUYGENS, and HOSTE among others it took until almost the mid-eighteenth century before a mature scientific theory of ship hydrostatic stability was formulated and published. Pierre BOUGUER (Bouguer 1746, “Traité du Navire”) and Leonhard EULER (Euler 1749, “Scientia Navalis”) were the founders of modern ship stability M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_8, © Springer Science+Business Media B.V. 2011

141

142 H. Nowacki and L.D. Ferreiro

theory, who quite independently and almost simultaneously arrived at their landmark results for hydrostatic stability criteria. BOUGUER developed the theory and introduced the terminology of the metacenter and the metacentric curve. EULER defined the criterion of the initial restoring moment, which for hydrostatic stability amounts to an equivalent concept. The full implementation of computational methods for evaluating these criteria and their acceptance by practitioners took even several decades longer. Paradoxically this chain of events raises two nearly contradictory questions, which this paper will address:  Why did it take so long for these formalized, quantitative criteria for the stability of ships of arbitrary shape to be pronounced?  When at last the discoveries were made, why then did two independent, but equivalent solutions suddenly appear almost at the same time? The knowledge required to evaluate the stability of a ship rests on many concepts and requires many autonomous discoveries to be made and to be brought into concerted application. These include:  The idea of conceptual experiments in dealing with mechanical systems.  The abstraction of thinking in terms of resulting forces and moments of weight and buoyancy (“lumped effects”) substituted for distributed effects of gravity and pressure.  The axiom of force equilibrium, here between weight and buoyancy force (Principle of ARCHIMEDES).  The axiom of moment equilibrium in a system at rest.  A definition and a test for system stability.  A method for finding the combined weight and center of gravity (CG) for several weight components.  A principle for finding the resultant buoyancy force and its line of action (through the center of buoyancy, CB).  A method for calculating volumes and their centroids, first for simple solids, then for arbitrary ship shapes.  An analytical formulation of a stability criterion (for infinitesimal and for finite angles of inclination). Although the physical principles of hydrostatic stability were already established by ARCHIMEDES, it took a long time before the analytical formulation of stability criteria could be pronounced for the general case of ships, essentially by means of calculus, and before numerical evaluations became feasible. BOUGUER and EULER were the first to find an analytical criterion for initial stability, BOUGUER in terms of the metacenter. EULER in terms of the initial restoring moment. BOUGUER went beyond this in several practical aspects. The intriguing question remains: How did two scientific minds work independently to come to rather equivalent results? What were their sources, their background, their approach, their logic, their justification and verification? Which were their unique original thoughts and how did they differ?

Historical Roots of the Theory of Hydrostatic Stability of Ships 143

To answer these questions it is not sufficient to look only at their final results and conclusions, but it is necessary to examine more closely the methodical approaches taken to the ship stability issue and to compare the trains of thought by which the individual authors arrived at their results. In this article we will review the developments that led to this historical stage when modern hydrostatic stability theory was founded.

2 Precursors 2.1 Archimedes ARCHIMEDES of Syracuse (ca. 287-212 B.C.), the eminent mathematician, mechanicist and engineering scientist in antiquity, is also the founder of ship hydrostatics and hydrostatic stability, which he established as scientific subjects on an axiomatic basis. ARCHIMEDES was brought up in the early Hellenistic era in the tradition of Greek philosophy, logical rigor and fundamental geometric thought. ARCHIMEDES was well educated in these subjects in Syracuse and very probably also spent an extended study period in Alexandria, the evolving Hellenistic center of science, at the Mouseion (founded in 286 B.C.). There he met many leading contemporary scientists, e.g., DOSITHEOS, ARISTARCHOS and ERATOSTHENES, with whom he maintained lifelong friendships and scientific correspondence. Steeped in this tradition of Greek geometry and mechanics, ARCHIMEDES learned how to excel in the art of deductive proofs from first premises usually based on conceptual models, i.e., models of thought, by which physical reality was idealized. But ARCHIMEDES was also unique, as many legends on his engineering accomplishments tell us, in applying practical observation to test his scientific hypotheses and to develop engineering applications, although these achievements are not mentioned in his own written work. We are fortunate that many, though not all, of ARCHIMEDES’ treatises are preserved, all derived from a few handwritten copies made in the Byzantine Empire during the 9th and 10th centuries and transmitted to posterity on circuitous routes to resurface essentially through the ARCHIMEDES revival during the Renaissance. Luckily, one Greek manuscript, which was later lost, had been translated into Latin by the Dominican monk Willem van MOERBEKE. This translation, which was published in 1269 and was later named Codex B, also contained ARCHIMEDES’ treatise “On Floating Bodies”, Books I and II. It became the only accessible reference to ARCHIMEDES’ work on hydrostatics for many centuries until in 1906 most surprisingly an old 10th century palimpsest was rediscovered in a Greek monastery in Constantinople by J. HEIBERG (Heiberg 1906-07). This palimpsest, called Codex C by HEIBERG, under the writing of a 12th century monk of a Greek prayerbook had retained significant traces of the

144 H. Nowacki and L.D. Ferreiro

rinsed off text of a Greek manuscript from ARCHIMEDES including the Greek version of “On Floating Bodies”. ARCHIMEDES’ texts were soon reconstructed from this source, transcribed and translated into modern languages (Archimedes 2002). Meanwhile this palimpsest, which had disappeared in private possession in the aftermath of the Greek-Turkish war in 1920-22, has turned up again recently and is under new scientific evaluation at the Walters Art Museum in Baltimore (Netz and Noel 2007). It is on these sources primarily that we can base a reliable evaluation of ARCHIMEDES’ contributions to ship hydrostatics today. ARCHIMEDES preceded his work on hydrostatics by a number of other treatises establishing certain axioms of mechanics: 2.1.1 The Law of the Lever: In his treatise “The Equilibrium of Planes” ARCHIMEDES first deals with the equilibrium of moments about a fulcrum in a lever system. Although he claims to deduce this principle from geometric reasoning alone, it is actually understood today that the law of the lever is equivalent to the axiom of moment equilibrium in mechanics. Second, he introduces the concept of “centroids” of quantities (areas, volumes, weights) into which the quantities can be “lumped” as concentrated effects so that moment equilibrium is retained. Third, he proposes a method for finding the “compound centroid” of a system of components, e.g., a center of gravity. Finally, he proves the critical “centroid shift theorem” i.e., a rule for the shift of the system centroid when some quantity is added to, removed from or shifted within the system. All of these concepts and results are essential physical principles as prerequisites for his work on hydrostatics. 2.1.2 Quadrature: In his treatise “The Quadrature of the Parabola” ARCHIMEDES illustrates by the example of the parabola how the Greeks determined areas and volumes of elementary shapes without the availability of calculus. Here he uses a method, well-known since EUDOXUS (Boyer 1939, 1949), based on an inscribed polygonal approximant which is continually refined by interval halving until under the given premises it converges to the given curve within a specified error tolerance. This type of deduction was later called “method of exhaustion”. Thus the quadrature problem is reduced to that of evaluating the sum of a finite, truncated or sometimes even infinite series of approximations. This method of quadrature generally is not equivalent to calculus for lack of a limiting process to infinitesimal step width, but has nonetheless inspired many future developments until this day.

Historical Roots of the Theory of Hydrostatic Stability of Ships 145

2.1.3 The Method of Mechanical Theorems: This famous treatise, which was actually also discovered in the palimpsest of 1906, explains how ARCHIMEDES used a reasoning based on principles of mechanics (like moment equilibrium of volume quantities) in deriving geometrical results (like volume centroids). He regarded such findings as propositions for later rigorous deduction by strictly geometrical proofs. 2.1.4 On Floating Bodies: In this treatise ARCHIMEDES makes use of all these prior results and proceeds to lay the foundations for ship hydrostatics and stability in the following steps (Archimedes 2002, Czwallina-Allenstein 1996): Book I of this treatise begins with Postulate 1 describing the properties of a fluid at rest axiomatically. “Let it be supposed that the fluid is of such character that, its parts lying evenly (N.B.: at the same level) and being continuous (N.B.: coherent), that part which is thrust the less is driven along by that which is thrust the more and that each of its parts is thrust by the fluid which is above it in a perpendicular direction, unless the fluid is constrained by a vessel or anything else”. Although ARCHIMEDES does not use the word “pressure” and the Greeks did not know that concept in antiquity, he does infer that parts under more pressure would drive parts under less pressure so that a fluid cannot be at rest unless the pressure is uniform at a given depth, while the weight of a vertical column of fluid rests on the parts below it. From these very simple axiomatic premises, which do not permit evaluating the local pressure anywhere in the fluid, ARCHIMEDES is able to derive the principles of hydrostatic equilibrium and stability of floating bodies. This is achieved by considering the equilibrium of the resultant buoyancy and gravity forces and of their moments. ARCHIMEDES’ famous Principle of Hydrostatics is stated in Book I, Proposition 5: “Any solid lighter than a fluid will, if placed in the fluid, be so far immersed that the weight of the solid will be equal to the weight of the fluid displaced” (Archimedes 2002). The proof of this law, usually pronounced today as Δ = γ V, is brilliantly brief and conclusive. In all brevity it rests on the argument that in equilibrium the solid is at rest in a fluid at rest, thus if the body is removed from the fluid and the cavity left by its underwater volume is filled with fluid matter, then the fluid can only remain at rest if the replacing fluid volume weighs as much as the solid, else the fluid would not remain in equilibrium and hence at rest.

146 H. Nowacki and L.D. Ferreiro

In Book II, mainly Proposition 2, ARCHIMEDES deals with the stability of hydrostatic equilibrium by treating the special case of a solid of simple shape, viz., a segment of a paraboloid of revolution of homogeneous material whose specific gravity is less than that of the fluid on whose top it floats. In equilibrium it floats in an upright condition. The stability is tested by inclining the solid by a finite angle to the vertical, but so that the base of the segment is not immersed. The equilibrium is defined as stable if the solid in the inclined position has a restoring moment tending to restore it to the upright condition. For the homogeneous solid this stability criterion is readily evaluated geometrically by examining the lever arm between the buoyancy and the gravity force resultants (Fig. 1). The buoyancy force acts through the centroid of the underwater volume B, which ARCHIMEDES finds for the inclined paraboloid from theorems proven earlier. The gravity force or weight acts through the Center of Gravity R of the homogeneous solid. Our conventional righting arm, the projection of BR on the horizontal, is positive. Instead of using this stability measure, ARCHIMEDES takes a shortcut for this homogeneous solid by splitting off and removing the weight of the submerged part of the solid Δ1 and the corresponding equal share of the buoyancy force, which have no moment about B because they both act through B. Thus only the weight of the abovewater section of the solid Δ2, acting through C, and the equal and opposite buoyancy force increment, acting through B, are taken into account. The centroid C is found from B and R by applying the centroid shift theorem when removing the underwater part from the system. This yields a positive “incremental righting arm” for the force couple of Δ2, acting through B and C respectively. The restoring moment is thus positive and the solid will return to the upright position. This application of the hydrostatic stability criterion by ARCHIMEDES is limited to the special case of a homogeneous solid of simple parabolical shape. It demonstrates the physical principles of the hydrostatic stability problem for a finite angle of inclination. It does not extend to floating bodies of arbitrary shape and of non-homogeneous weight distribution, hence actual ships. Still the foundations were laid to enable others much later to treat the generalized case of the ship on the same fundamental grounds. A more detailed appraisal of ARCHIMEDES’ contributions to the hydrostatic stability of floating systems and a thorough historical account is given by NOWACKI (Nowacki 2002).

Historical Roots of the Theory of Hydrostatic Stability of Ships 147

Fig. 1 Restoring moments, righting arms for inclined homogeneous paraboloid, based on ARCHIMEDES’ “On Floating Bodies”, Book II (Archimedes 2002) (figure from (Nowacki 2002)).

2.2 Stevin and Pascal In the beginning of the modern era of science the manuscripts of ARCHIMEDES had been rediscovered by several humanists during the Renaissance and were made accessible in print after 1500 by several editions in Greek, Latin and later by translations into modern languages (Archimedes 2002, Czwallina-Allenstein 1996, Dijksterhuis 1987). Simon STEVIN (born 1548 in Bruges, died 1620 in Leyden), the celebrated Flemish/Dutch mechanicist and hydraulic engineer, who knew and admired ARCHIMEDES’ works, was probably the first modern scientist who resumed and resurrected the study of hydrostatics and applied it to hydraulics, but also to ships. In his important work “De Beghinselen des Waterwichts” (Stevin 1955), first published in Dutch in 1586 and translated into Latin by Willibrord Snellius in 1605, he deals with the principles of hydrostatics and hydraulics. He adopts the idea of “specific gravity” of a fluid from ARCHIMEDES and introduces the concept of “a hydrostatic pressure distribution”, as we would call it today, proportional to the weight of a water prism reaching down to the depth in question. This enables him to calculate water loads on walls in a fluid, an important foundation in hydraulic engineering. He also rederives ARCHIMEDES’ Principle of Hydrostatics. However when he proceeds to examine the hydrostatic stability of a ship in his later note “Van de Vlietende Topswaerheyt” (Stevin 1586), he correctly reconfirms that for equilibrium buoyancy and weight force resultants must act in the same

148 H. Nowacki and L.D. Ferreiro

vertical line through the centers of buoyancy (CB) and gravity (CG). But he erroneously concludes that for stability the CB must always lie above the CG. This error occurred because – unlike ARCHIMEDES - he neglected the centroid shifts resulting from the volume displacement from the emerging to the immersing side in heel. Despite this flaw in an application STEVIN deserves high recognition for founding modern hydrostatics on the concept of hydrostatic pressure. Blaise PASCAL (born 1623 in Clermont, died 1662 in Paris) is also counted among the founders of modern hydrostatics. He was familiar with ARCHIMEDES’ and probably with STEVIN’s work. In his “Traité de l’ équilibre des liqueurs”, Paris (1663), he arrives at similar conclusions and justifications regarding the fundamentals of hydrostatics in a fluid as STEVIN did, though he did not deal with ships. The assertion and experimental verification of hydrostatic laws also being applicable to air belong to his original contributions to this subject.

2.3 Huygens Christiaan HUYGENS (born 1629 in The Hague, died 1695 in The Hague), the eminent Dutch physicist, at the youthful age of 21 made an excursion into hydrostatic stability, which is not well known. In 1650 he wrote a three volume treatise “De iis quae liquido supernatant” (Huygens 1967a), in which he applies the methodology of ARCHIMEDES to the stability of floating homogeneous solids of simple shape, reconfirming ARCHIMEDES’ results and extending the applications to floating cones, parallelepipeds, cylinders etc., at the same time studying the stability of these solids through a full circle of rotation. He never published this work in his lifetime because he felt it was incomplete or “of small usefulness if any” or in any case not sufficiently original in comparison to ARCHIMEDES. He wanted the manuscript to be burnt, but it was found in his legacy and at last published in 1908. HUYGENS must be admired for his deep insights into ARCHIMEDES’ fundamentals of hydrostatics and for his own creative extensions. He recognized e.g. that for homogeneous, prismatic solids their specific weight and their aspect ratio are the essential parameters of hydrostatic stability. In his derivations he used a formulation based on virtual work as a principle of equilibrium. These examples of famous physicists between 1500 and 1700 working in hydrostatics and on hydrostatic stability of floating solids and ships demonstrate that the foundations inherited from ARCHIMEDES were understood by certain specialists, but had not yet been extended or applied to the design and stability evaluation of actual ships. Although the physical principles of stability were understood, the practical evaluation of volumes, volume centroids, weights and centers of gravity for floating bodies of arbitrary shape and non-homogeneous weight distribution de facto still posed substantial difficulties. These were not overcome before the advent and application of calculus during the 18th century.

Historical Roots of the Theory of Hydrostatic Stability of Ships 149

2.4 Hoste The French mathematics professor Paul HOSTE, s.j., (1652-1700) was the first to attempt to quantify the problem of ship stability. He did so in his 1697 treatise of naval architecture, Théorie de la construction des vaisseaux, which was appended to his book on naval tactics, written at the behest of Admiral TOURVILLE (Hoste 1697). However, he did not apply calculus to the problem, which was not yet widely known. HOSTE assumes, without citing STEVIN, that the CG and CB were in a vertical line; but he also allows without proof that the CG could be above the CB. However, to explain how this could be possible without the ship tipping over, he also assumes that the buoyancy force was equally divided between the two halves of the ship, forming the base of a triangle. He continues this error by stating: “If the center of gravity of the ship is known, the force with which it has to carry sail is easily known, which is no other thing than the product of the weight of the ship by the distance between these centers (of weight and displacement)” (Hoste 1697, p.54). In modern terms, the righting moment is equal to Δ x ( KG – KB ). In other words, the higher the center of gravity, the more stable the ship. Although he does not provide a theoretical means for determining this “power to carry sail”, HOSTE does describe a procedure which could empirically demonstrate this: The inclining experiment. HOSTE asserts that, by measuring the angle of inclination due to suspending a weight from a boom at a certain height, that the “force to carry sail” can be determined (Hoste 1697, p. 56). Although HOSTE’s argument contains several fundamental errors, it was the first attempt to express the stability of a ship in mathematical terms, and his book remained the only published inquiry into stability for almost half a century.

2.5 La Croix César Marie de LA CROIX (1690-1747) was the head of administration and finance for the Rochefort dockyard, and maintained the records for the galley fleet. He was not a scientist or engineer. Yet he developed some fundamental concepts on ship motions and hydrostatic stability, derived for a floating parallelepiped (La Croix 1736), so certainly before BOUGUER’s and EULER’s work appeared. LA CROIX was interested in the motions, but also in the hydrostatic restoring moments for this parallelepiped when heeled. He correctly understood the role of weight and buoyancy forces acting in the same vertical line in equilibrium. He also followed ARCHIMEDES in his stability criterion by examining the heeled body and requiring positive righting arms, accounting for the wedge volume shifts. But he falsely determined the influence of the wedge shift moments and

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hence the righting arms. Besides, since he lacked integration methods based on calculus, he was not able to generalize his results for arbitrary section and waterplane shapes, hence to make them applicable to ships. Yet although LA CROIX’s work was flawed, it may have triggered EULER’s renewed interest in ship hydrostatics. In 1735 EULER was asked by the Russian Imperial Academy of Sciences to review the treatise (La Croix 1736), which LA CROIX had submitted there prior to publication. EULER quickly responded and appraised the merits of LA CROIX ’s problem formulation, but also pointed out the weaknesses of the solution. He then put on record his own correct solution for the initial restoring moment, hence stability criterion, for the parallelepiped (or any other body of constant cross sectional shape), which he claimed to have found earlier (Euler 1972). This dates EULER’s earliest written mention of the criterion for this special case to 1735. In 1736 in reply to LA CROIX ’s rebuttal EULER also provided the complete explicit derivation of this result.

2.6 Sailing Vessel Propulsion and Maneuvering Questions of ship propulsion, ship motions and maneuvering, but also of ship stability have always aroused an acute interest, not only among seafarers and shipbuilders, but also in the scientific community. The propulsion of sailing vessels by the forces of wind, e.g., was investigated by ARISTOTLE in antiquity and later is associated with such celebrated names of scientists as Francis BACON, Thomas HOBBES and Robert HOOKE, among others. Toward the end of the 17th century the subject of the propulsion of sailing vessels had gained new scientific interest, also under the influence of the navies in applying scientific principles to ship design and operation. Treatises and monographs on the theory of maneuvering and on the equilibrium of aerodynamic propelling forces and hydromechanic response had appeared, notably by IgnaceGaston PARDIES (1673), Bernard RENAU d’ELISSAGARAY (1689), Christiaan HUYGENS (1693), Jakob BERNOULLI (1695) and Johann (I) BERNOULLI (1714). This had led to lively controversies, but also to an increasing depth of understanding of the mechanics of the sailing vessel. In fact through the various treatises the principles for applying hydrodynamic underwater and aerodynamic sail forces to the vessel in order to establish its equilibrium position had become well understood. At the same time the importance of allowing for trim and heel (as well as yaw) under these forces and moments had been recognized and dealt with, although the quantitative assessment of the hydrostatic restoring effects remained largely an open issue. Both BOUGUER and EULER would begin their investigations into the nature of hydrostatic stability with the study of propulsion and maneuvering of sailing ships.

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2.7 Displacement Estimates Before Calculus The infinitesimal calculus was important to the development of stability for several reasons, including the practical evaluation of underwater volume and volume centroids. Yet a number of methods to calculate underwater volumes (and therefore displacements) were developed in the 1500s and 1600s, well before any theoretical framework for stability was available to make use of it. Why would shipbuilders go through the trouble of making such calculations? It appears that there were two separate reasons for this: The development of the gunport, and the measurement of cargo tonnage. The gunport was introduced in the early 1500s, and came into wide use by the middle of the century. This greatly increased the firepower of naval ships, but brought two problems; ships got much heavier, and with large holes in the side of the ship, the available freeboard got dramatically smaller. Thus, the margin for error in estimating the waterline was considerably reduced. To handle this problem, shipbuilders like Anthony DEANE (1638-1721) developed methods to calculate how much armament, ballast and stores should be loaded on a ship to bring it to the correct waterline below the gunports. DEANE and others made use of ship’s plans, which were just beginning to be employed by shipbuilders as a construction guide. In his manuscript “Doctrine of Naval Architecture” (Deane 1981), never published but widely circulated, DEANE demonstrates two methods to calculate the area underneath waterlines at each “bend” or frame of the hull; using either (1) an approximation for the area of a quarter-circle or (2) by dividing the area into rectangles and triangles. DEANE then sums the areas for each frame, multiplies by the frame spacing and multiplies the volume by the specific weight of water to obtain the displacement. He does this for several different waterlines, including the desired waterline below the gunports. When a ship is launched, he can immediately determine the light displacement, and then calculate how much weight should be added to arrive at the design waterline. A second reason for introducing displacement calculations was to more accurately measure the cargo capacity of a ship. For example, from 1646 to 1669 the Dutch and Danish governments carried on a series of negotiations on cargo measurement. In 1652 a Dutch mathematician Johannes HUDDE identified the basic issue of measuring cargo deadweight by determining the difference between the weight of the ship empty and fully-laden, using a difference-in-drafts method. He suggested to the Dutch authorities that they measure the waterplane areas at each draft of an actual ship in the water (not from plans), by taking measurements to the hull from a line extended at the side of the ship parallel to the centerline. The space between hull and an overall rectangle formed by the length and beam was then divided into trapezoids and triangles, the areas calculated and summed, multiplied by the difference in drafts and multiplied by seawater density, to obtain cargo tonnage. Although the suggestion was never used, HUDDE’s cousin

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Nicolas WITSEN reported it in his 1671 shipbuilding manuscript “Aeloude en hedendaegsche scheepsbouw en bestier” (Witsen 1671; Cerulus 2001).

3 Development of Stability Theory 3.1 Bouguer 3.1.1 Biographical Sketch Pierre BOUGUER (Fig. 2) was born on 10 February 1698 in the French coastal town of Le Croisic, near Saint Nazaire at the mouth of the Loire. Educated in the Jesuit school in Vannes, he quickly showed himself a child prodigy. When his father Jean BOUGUER, a royal hydrographer and mathematician, died when Pierre was only 15, he applied for his father’s position. After initial hesitation by the authorities, he passed the rigorous exam and was given the post of Royal Professor of Hydrography. BOUGUER won several Royal Academy of Sciences prizes for masting, navigation and astronomy before he was 30. He moved to the port of Le Havre in 1731, about the same time he became a member of the Academy. His work caught the attention of the French Minister of the Navy MAUREPAS (1701-1781), who, like COLBERT before him, was convinced of the strategic benefit of ship theory as a way of compensating by quality against the quantitative superiority of the British navy. MAUREPAS supported BOUGUER’s research and sponsored his publications. In 1734, BOUGUER became involved in a controversy over the Earth’s shape; those who believed in DESCARTES’ vortex theories of physics held that the Earth was elongated at the poles, while those who accepted NEWTON’s theories of gravitational attraction argued that the Earth was wider at the equator due to centrifugal force. MAUREPAS also held the view that a full understanding of the Earth’s shape was essential to navigation, so he sent BOUGUER, along with several other members of the Academy of Sciences (accompanied by two Spanish naval officers) on a Geodesic Mission to Peru to determine the length of a meridian arc for a degree of latitude at the equator. One of his companions on the Mission was the young Spanish lieutenant Jorge JUAN Y SANTACILIA, who would later become a prominent naval constructor and author of a highly recognized treatise on naval architecture. BOUGUER spent ten years away (1735-1744), during which time he not only surveyed and calculated a meridian arc length of three degrees of latitude, he also performed various experiments on gravity and astronomy. It was during this Mission, in the peaks and valleys of the Andes far from the ocean, that he wrote much of his monumental work “Traité du navire”, the first comprehensive synthesis of naval architecture.

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Fig. 2 Portrait of Pierre BOUGUER.

“Traité du navire” was published in 1746, shortly after BOUGUER’s return. He remained unmarried, lived in Paris and devoted himself to revising his geodesic work, and carrying out further studies of naval architecture, astronomy, optics, photometry and navigation. BOUGUER died in Paris on 15 August 1758, aged 60. Further details on BOUGUER’s biography and scientific work are presented by FERREIRO (Ferreiro 2007) and DHOMBRES (Dhombres 1999). 3.1.2 Early Work on Ship Theory In 1721 BOUGUER was asked by the Academy of Sciences in Paris to compare the accuracy of two methods of calculating cargo tonnage being proposed to the Council of the Navy for port fees. Although the mathematician Pierre VARIGNON proposed estimating the volume of a ship as a semi-ellipsoid, BOUGUER found that the best results came from a proposal of Jean-Hyacinthe HOCQUART of

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Toulon, who used a difference-of-waterplanes method that employed equal-width trapezoids to estimate waterplane areas. This was similar to HUDDE’s approach but allowed a direct calculation of areas from ship’s plans. BOUGUER would later adopt and refine this “method of trapezoids” for his own stability work (Mairan 1724). In 1727 the Academy offered a Prize for the best treatise on masting, which BOUGUER’s entry won. He postulated that a “point vélique”, the intersection of the sail force and the resistance of water against the bow, should be directly above the center of gravity to minimize trimming by the bow. In his treatise he relied on HOSTE’s theories to explain stability, although where HOSTE implied that the advantage of doubling is through an increase in the center of gravity, BOUGUER invoked ARCHIMEDES to point out that the buoyancy of the added portion of the ship moves the center of buoyancy laterally when heeling, thus increasing the righting moment. Still, this treatise did not yet show any insights into the evaluation of trim or heel angles or restoring moments that he would develop five years later (Bouguer 1727). 3.1.3 The Metacenter BOUGUER probably began formulating his theory of stability around 1732, after he had moved to Le Havre, for he tested it using the little 18-gun frigate Gazelle, laid down in that dockyard in May 1732 and delivered in January 1734 (Demerliac 1995). However, no letters or manuscripts from that period survive to confirm this, and his derivation of the metacenter would not appear in print until 14 years later in “Traité du navire”. The following steps illustrate BOUGUER’s derivation of the metacenter in his “Traité du navire” (Bouguer 1746, pp. 199-324). STEP 1: Premises and axioms BOUGUER implicitly defines the hydrostatic properties of fluids, based on the principle of hydrostatics that weight and buoyancy are equal, opposite and act in the same vertical line. He does so without proof, without the use of equations and without mentioning ARCHIMEDES by name, although he shows general familiarity with his work. He also implies through geometrical arguments that the pressure of the fluid follows a hydrostatic distribution with depth and is everywhere normal to the surface of the submerged body.

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Fig. 3 BOUGUER’s diagram of the metacenter (Bouguer 1746, fig. 54).

He denotes: p = submerged volume Γ = upright CB γ = inclined CB AB = upright waterline ab = inclined waterline I = CG of an unstable ship G = CG of a stable ship g = metacenter 1 = centroid of immersed wedge 2 = centroid of emerged wedge 3 = centroid of body without wedges STEP 2: Magnitude of buoyancy force BOUGUER resolves the submerged surface into very small elements (though without using any calculus notations at this point) and equates the vertical components of the hydrostatic pressure forces to the weight of the water column resting on top of the element in the interior of the submerged volume conceptually. Hence, the total pressure resultant is equal to the total weight of water filling the submerged volume. He thus reconfirms the law of ARCHIMEDES by pressure integration. STEP 3: Measurement of volume and centroids BOUGUER first suggests two methods for calculating the volume of the ship as a regular solid. The first is to model the ship as an ellipsoid, as originally proposed by VARIGNON, and the second is to divide it into prisms. The set of quadrilateral prisms then forms a polyhedral approximant of the hull surface for volume summation. BOUGUER sees clearly that, by analogy, the same principle of

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polygonal approximation also holds for evaluating the area of planar curves. He thus quickly homes in on a quadrature rule for curves based on equally wide trapezoids, his favorite rule, also known as trapezoidal rule. He uses it to measure areas of waterline “slices” and then combining those two-dimensional slices to obtain a three-dimensional volume. Although he borrowed the idea from HOCQUART ’s 1717 proposal, BOUGUER refined the method, first by dividing each waterplane into many sections (HOCQUART only proposed three sections), and second by taking the areas of several waterlines to develop the entire volume of the hull (HOCQUART took only one “slice”). BOUGUER then arrives at the conclusion that his quadrature rule is suitable for evaluating the integral of any continuous function of one variable or, if applied recursively, for continuous functions of any number of independent variables. Interestingly he thus interprets the analytical formulation of stability criteria, which he introduces later, as being equivalent to the discretized quadrature rules presented here earlier. The ancestry of his concepts from both practical shipbuilding traditions and modern calculus is still leaving some visible traces. BOUGUER then digresses for many pages into using the trapezoidal rule to calculate incremental waterlines for estimating a ship’s payload capacity, and gives various rules for tunnage admeasurement. For finding the centroid of the underwater volume, the center of buoyancy CB, BOUGUER then begins by explaining, in very simplistic terms, how to use the sum-of-moments method to determine the centroid of an object. He then derives the area centroid of a planar figure (2D case), then the volume centroid of a solid (3D case), which for a ship he calls the “center of gravity of the hull”. He then discusses by example how to evaluate these expressions numerically by the method of trapezoids. STEP 4: Stability criterion The center of buoyancy having been determined, BOUGUER next explains why he chooses the metacenter as the initial stability criterion, using the geometrical argument shown in Fig. 3. The ship’s center of gravity g is always in the same vertical line as the center of buoyancy Γ, but this geometry is not constant due to the ship’s movement. If the ship has a very high center of gravity I, and moves even a little from the upright position (waterline A-B) to another position (waterline a-b), it is no longer statically stable; the center of buoyancy moves from Γ to γ, i.e., away from the vertical of the center of gravity, and the vertical force of buoyancy shifts from Γ-Z to γ-z. The ship’s weight, centered at I and on the opposite side of the inclination from the new center of buoyancy, tends to push the ship even further over, rendering it unstable. However, if the center of gravity of the ship is at G, below the intersection g of the upright and inclined vertical forces of buoyancy, then the center of gravity is on the same side as the new center of buoyancy and the resulting force always tends to restore the ship to the horizontal. BOUGUER states in his definition of the metacenter (Bouguer 1746, pp. 256257):

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“Thus one sees how important it is to know the point of intersection g, which at the same time it serves to give a limit to the height which one can give the center of gravity G, it [also] determines the case where the ship maintains its horizontal situation from that where it overturns even in the harbor without being able to sustain itself a single instant. The point g, which one can justly title the metacenter [BOUGUER’s italics] is the term that the height of the center of gravity cannot pass, nor even attain; for if the center of gravity G is at g, the ship will not assume a horizontal position rather than the inclined one; the two positions are then equally indifferent to it: and it will consequently be incapable of righting itself, whenever some outside cause makes it heel over.” BOUGUER does not use the terms “stable” or “unstable”, but rather states that G is either lower or higher than g. Step 5: Evaluation of the criterion The determination of this point of intersection, BOUGUER says, reduces to the question of the distance between the centers of buoyancy Γ and γ of the submerged body upright and just slightly inclined. BOUGUER employed the allimportant shifting of equal immersed and emerged wedges to determine this distance, by drawing a triangle between points 1, 2 and 3. Since the centers of buoyancy Γ and γ lie on the legs of that triangle, the distance Γ – γ must be proportional to the distance between points 1 and 2, and the distance between Γ and point 3 must be proportional to the ratio of the volume of the underwater hull and the wedges. This geometrical explanation sets the stage for the second half of the explanation, the mathematical analysis of the mechanics of stability. Step 6: Results In the next section, BOUGUER uses calculus to determine the three unknowns: The distance from point 1 to point 2; the volume of the wedges; and the volume of the hull. BOUGUER imagines that Fig. 3 represents the largest section of the ship, which actually extends through the plane of the page in the longitudinal direction x, with the immersed and emerged wedges actually an infinite sequence of triangles of width y (the largest being b = F-B, or the half-breadth of the hull at the waterline) and height e ( = H-B) going through the length of the hull at a distance dx from each other; integrating, the volume of the wedge is Vwedge =

e ∫ y2 dx 2b

(1)

The second unknown is the distance between the centers of the wedges, points 1 to 2; but since the center of a triangle is two-thirds the height from apex to base, it is straightforward to obtain this distance as

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3 Distance 1- 2 = 4  y dx 3  y 2 dx

(2)

The third unknown, the volume of the hull p, is derived using the trapezoidal rule. Putting the three together, the distance is:

3 Γγ = 2 e  y dx 3 bp

(3)

Finally, observing that the triangle Γgγ (i.e., between the centers of buoyancy and the intersection of their vertical lines of force) is similar to the triangles formed by the immersed and emerged wedges, the height of the metacenter g above the center of buoyancy Γ is found via Euclid to be: 3 Γg = 2  y dx 3p

(4)

This is the now-famous equation for the height of the metacenter. 3.1.4 Implications of the Metacenter BOUGUER develops in thorough detail both the theoretical and practical aspects of the metacenter. He derives the metacenter for various solids (ellipsoid, parallelepiped, prismatic body) and presents the procedures for its practical, numerical calculation for ships. These explanations were detailed enough for practical applications and became the foundation for later textbooks. But BOUGUER also charged ahead beyond the initial metacenter for infinitesimal angles of heel when he introduced the concept of the “métacentrique”, i.e., the metacentric curve for finite angles of heel. First, he clearly recognized that the same physical principles and stability criteria apply to an inclined position of the ship as they do for the upright case. Second, he recognized that the metacentric curve for finite angles is in fact the locus of the centers of curvature of the curve of the centers of buoyancy. This was a brilliant, original insight. The locus of the centers of curvature of a curve was known since Christiaan HUYGENS’s work on the pendulum clock (Huygens 1967b) under the name of “développée” (or evolute). Third, BOUGUER also knew how to construct the evolute of a given curve. For a wall-sided ship (or parallelepiped) the metacentric curve is a cusp shaped curve composed of two hyperbolas lying above the metacenter, as BOUGUER showed. His demonstration examples for the “métacentrique” do document that he understood

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how to approach stability for finite angles of heel, though he never used a “righting arm” criterion. On practical aspects of initial stability he recommends that the widest point of the ship be no lower than where the maximum heel would be before it begins to tumble-home, or even that ship sides be straight or flared throughout. In other words, BOUGUER was advising to avoid tumble-home altogether, although in somewhat oblique terms. BOUGUER next provides a numerically-worked example of the application of the metacenter to a real ship, underlining the importance of Gg, the distance between the center of gravity and the metacenter. Using the Gazelle as his model, BOUGUER explains how to account for the weight of each part of the ship, including the frames, planking, nails, etc., how to measure the center of gravity of each piece using the keel as the reference point, and how to sum the moments to obtain the overall center of gravity G. He then calculates displacement and center of buoyancy of the ship using the trapezoidal rule, which enables him to find the metacenter g. His calculations confirmed that, once properly ballasted, the Gazelle would be stable. BOUGUER continues for almost 50 pages to outline the practical implications of the metacenter on hull design and outfitting. In many cases the implications are re-statements of what constructors already knew; but BOUGUER provided for the first time a rigorous analysis of why they were true. Some of the main points he brings out are:  The greatest advantage to stability lies in increasing the beam. BOUGUER claims that the “stability” varies as the cube of the beam, by which he must have been referring to the transverse moment of inertia I ,T while the metacentric height requires closer scrutiny. But he wanted to emphasize the rapid increase of stability with beam. This also explains for the first time why doubling a ship improves stability, although BOUGUER does not explicitly state so.  Stability is improved by diminishing the weight of the topsides; though this was well-understood, BOUGUER details how to accurately assess the effects. BOUGUER also correctly describes for the first time how to evaluate the inclining experiment, for whose basic idea he credits HOSTE, in order to determine Gg, the distance between the center of gravity and the metacenter. In Fig. 4, he demonstrates how to use a known weight suspended from a mast to incline the ship and measure the angle of heel. Using the law of similar triangles, he shows that the ratio of the ship’s displacement and the suspended weight is proportional to the ratio of Gg to the angle of heel and distance the weight moves; on the assumption of small angles of heel this allows Gg (i.e., the reserve of stability) to be calculated directly without knowing the exact position of the metacenter. BOUGUER demonstrates how to use his metacenter by detailing the calculations of weights and centers of gravity for Gazelle while it was still being framed out. It is therefore curious that he did not verify this by performing an inclining experiment on Gazelle. It would appear that, while BOUGUER was completing his initial stability work in 1734, he was caught up in the events that led to the

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Geodesic Mission, and would not return to the subject until he was in the mountains of Peru.

Fig. 4 BOUGUER’s diagram of an inclining experiment, (Bouguer 1746, fig. 55).

3.2 Euler 3.2.1 Biographical Sketch Leonhard EULER (Fig. 5) was born in Basel, Switzerland on 15 April 1707. He was the son of Paulus EULER, parson of the Reformed Church, and his wife Margaretha née BRUCKER. He went to a Latin grammar school in Basel and, as his father recognized his talent early, took private lessons in mathematics. In 1720 he enrolled at the University of Basel as a student and later as a young scientist, in the first three years in philosophy where he received a Magister’s degree, then in theology. But he soon turned his main interest to mathematics and mechanics, which he studied under Johann (I) BERNOULLI, who was recognized as a leading mathematician of that era. In fact Johann (I) BERNOULLI, who was 40 years EULER’s senior, saw EULER’s maturing genius and invited him to join the Saturday afternoon “privatissimum” in mathematics, a private circle held in the BERNOULLI’s home, where he also met and made friends with younger members of the BERNOULLI family, notably Niklaus (II), Daniel (I) and Johann (II) BERNOULLI. In this period Johann (I) BERNOULLI laid the foundations from which EULER’s later mathematical fame developed. In 1727 EULER, who had been looking for an academic position, received an offer from the Russian Imperial Academy of Sciences in St. Petersburg, which Tsar Peter had initiated and which had been opened in 1725, where Niklaus (II) and Daniel (I) BERNOULLI held appointments as professors from this beginning. EULER accepted this prestigious offer, starting as an Adjunct (élève) for a modest

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salary, and arrived there – after a three month voyage by coach and by ship – in June 1727. EULER began his scientific career in St. Petersburg very successfully, advanced to a professorship in physics and full membership in the Academy in 1731, and spent his “First St. Petersburg Period” (1727-1741) very productively, working on a wide range of scientific subjects and publishing more than 50 treatises and books. In 1741, amidst political transitions and uncertainties in Russia, EULER received an attractive offer from King FREDERICK II of Prussia to come to Berlin and to work for the Royal Academy of Sciences, which was to be founded. EULER moved there in 1741, continued his illustrious scientific work on an increasing variety of subjects, and when the Academy at last opened in 1746 soon became recognized as one of its most eminent members and as the leader of the mathematical class. He remained immensely productive and published some of his most famous books and treatises during his period in Berlin from 1741 to 1766. Despite EULER’s scientific fame and the considerable merits he earned in the Berlin Academy during its formative first two decades he never developed a relationship with King FREDERICK that made him feel recognized, appreciated and understood. This was one contributing factor why after 25 years in Prussian service he decided to return to St. Petersburg. There he was welcomed and honored at all levels, and spent his “Second St. Petersburg Period (1766-1783)” in relative comfort, peace and recognition despite his weakening health and fading eyesight. His scientific creativity never ceased. A large share of his more than 800 treatises and books also stems from this second period in Russia. His work encompasses the full gamut of scientific topics in his era, not only in mathematics and mechanics, but also in other branches of physics, astronomy, ballistics, music theory, philosophy and theology. He died in St. Petersburg in 1783. More detailed accounts of EULER’s life and scientific vita are given by BURCKHARDT et al. (Burckhardt et al 1983), FELLMANN (Fellmann 1995) and CALINGER (Calinger 1996). 3.2.2 Early Work in Ship Theory In 1726 the Académie des Sciences in Paris invited contributions to a prize competition on the optimum placement, number and height of masts in a sail propelled vessel. EULER as a student and protégé of Johann (I) BERNOULLI was of course familiar with the earlier treatises and disputes that had arisen about the forces acting on a sail propelled maneuvering vessel between RENAU, HUYGENS, Johann (I) BERNOULLI a.o. So he felt well enough prepared, also encouraged by Johann (I) BERNOULLI, to submit a treatise (“De Problemate Nautico…”) (Euler 1974) in 1727, which gained honorable mention, an “accessit” equivalent to a second prize. Pierre BOUGUER was awarded the first prize. In this treatise EULER, who still treads in the foot-steps of Johann (I) BERNOULLI, does not yet show deeper insights into ship hydrostatics although he does draw

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attention to the requirement that for a ship before the wind the propelling sail force is limited by the acceptable forward trim angle. But he has no reliable physical basis for estimating that angle. Yet this early experience in his life at the age of 20 gave EULER a first thorough acquaintance with the mechanics of ships, familiarized him with nautical and ship construction terminology and created in him a lasting interest in ships as topics to be studied by methods of mathematical analysis and engineering mechanics.

Fig. 5 Portrait of Leonhard EULER.

In St. Petersburg EULER soon had ample opportunities to return to these subjects. It is not clear when he first occupied himself more deeply with ship hydrostatics. But evidently the appearance of LA CROIX ’s treatise on the hydrostatic stability of a parallelepiped in 1735 found EULER well prepared, when the Academy asked for his review, to detect certain flaws in LA CROIX ’s derivations and to present the correct result for this simple shape, which first established EULER’s initial restoring moment criterion on hydrostatic stability. In his second review in 1736, in response to LA CROIX ’s rebuttal, EULER went beyond his first results, dealt with some other prismatic shapes (trapezoidal and triangular prisms) and

Historical Roots of the Theory of Hydrostatic Stability of Ships 163

alluded to his general approach to stability (Euler1972). This evidence is sufficient to date EULER’s earliest written mention of the hydrostatic stability criterion. There is also some indirect evidence of EULER’s earliest results on hydrostatic stability in his correspondence with the Danish naval constructor and naval attaché in London, Friderich WEGERSLØFF, which he maintained between 1735 and 1740. In a letter of 14 September 1736 WEGERSLØFF acknowledges receipt of a solution for the hydrostatic stability problem, on 22 May 1738 EULER replies and gives a derivation of his stability results and also mentions some experimental verification (Euler 1963). It is not clear from the context what sort of tests he had performed. By 1737 EULER’s interests in ship theory were well known at the Imperial Academy. Thus - probably not without his own prior knowledge or suggestion - he was commissioned by the Academy to write a book on this subject, resulting in his monumental two-volume “Scientia Navalis”. This work encompasses a presentation of the complete scope of ship theory according to the state of the art at that time and includes many new findings and derivations by EULER. The chapters in the two parts deal with ship hydrostatics, ship resistance and propulsion, maneuvering and ship motions. Many results have been of lasting value and have served as foundations for the growing scientific body of ship hydromechanics. According to EULER’s own report in the Preface of “Scientia Navalis”, he had worked on this manuscript from 1737 to 1740 (Euler 1967, pp. 15/16). When he left St. Petersburg in 1741 he had completed the first part, which contains all four chapters on hydrostatic equilibrium and stability, and half of the second part. The remaining sections of Part 2 were finished by 1741 in Berlin. Unfortunately, he had much difficulty finding a publisher for this voluminous opus, which thus was not published until 1749 by the Academy in St. Petersburg. Comprehensive appraisals of EULER’s overall contributions to ship theory and related matters are given by HABICHT (Habicht 1974), MIKHAILOV (Mikhailov 1983) and TRUESDELL (Truesdell 1983). 3.2.3 The Initial Stability Criterion To understand the history of EULER’s involvement in ship theory it is important to read the Preface to his “Scientia Navalis”. Here he explains that his work will go beyond the established disciplines of hydrography or nautical science and will concentrate on a physical and analytical investigation of the mechanics of ships, at rest and in motion, for which fundamental theoretical works were still missing at that time. In hydrostatics EULER departs from the Principle of ARCHIMEDES, to whom in the preface of “Scientia Navalis” he gives credit and praise. But he adds that the hydrostatic stability of ships must be newly approached and quantified in order to be able to distinguish between stable and unstable equilibrium of ships at the design stage. The experience of naval architects (“architecti navales”) alone, long

164 H. Nowacki and L.D. Ferreiro

established as it may be, will not be sufficient to prevent unexpected stability accidents. EULER acknowledges the motivation he received from reviewing LA CROIX’s treatise in 1735-1736, which prompted him to investigate more profoundly the transverse and longitudinal stability of ships. EULER also gives very favorable credit to BOUGUER’s “Traité du Navire”, which had appeared 3 years before “Scientia Navalis”, but he takes much care also to avert any suspicion of plagiarism by calling on the Imperial Academy as witnesses that he had written “Scientia Navalis” essentially between 1737 and 1741 during which period there was no communication with BOUGUER who was in Peru. These statements were never disputed between the two authors, who corresponded in a respectful and amicable fashion on other subjects later. EULER’s derivation of his stability criterion in “Scientia Navalis” proceeded in the following steps (Euler 1967, pp. 3-166): STEP 1: Premises and axioms In the first chapter of Book I, EULER deals with the equilibrium of bodies floating in water at rest. In essence he rederives the Hydrostatic Principle of ARCHIMEDES from the modern viewpoint of infinitesimal calculus by integration of the hydrostatic pressure distribution prevailing in a fluid over the surface of the body. The properties of the fluid at rest and the use of calculus for this purpose both were new at the time when EULER wrote these passages. As for the pressure distribution he states axiomatically in the opening paragraph of his book: “Lemma: The pressure which the water exerts on the individual points of a submerged body is normal to the body surface; and the force which any surface element sustains is equal to the weight of a straight cylinder of water whose base is equal to the same surface element and whose height is equal to the depth of the element under the water surface”. These brief axioms, viz. the normality of pressure to a surface and the inferred equality of the pressure at a point in a given depth in all directions, were the first analytical formulation for the properties of the fluid and are regarded as the necessary and sufficient conditions for the foundation of hydrostatics (Truesdell 1954). STEP 2: Magnitude of buoyancy force From these premises EULER proceeds to define by integral calculus the buoyancy force as the pressure resultant and the center of buoyancy (EULER calls the CB: “centrum magnitudinis”), through which it acts, as the volume centroid of the submerged part. He reconfirms also that for equilibrium the buoyancy and weight forces must act in the same vertical line and must be equal in magnitude and opposite in direction. He then illustrates these principles by examples of simple shapes like parallelepipeds and prisms of triangular and trapezoidal cross section. For each of these solids he finds the possible equilibrium conditions over a full circle of rotation as a function of the specific weight of these homogeneous solids, not unlike HUYGENS in his unpublished treatise of 1650.

Historical Roots of the Theory of Hydrostatic Stability of Ships 165

In Chapter II, briefly digressing from hydrostatics, EULER discusses the resulting motions of a floating body if it is temporarily displaced from its “upright” equilibrium position. Here he explains the “lumping” of masses in their center of gravity (CG), introduces the definition of the principal axes of inertia for ships, underscores the significance of the CG as a reference point, e.g., for decomposing a resulting motion into the translation of the CG and the rotation about it. EULER will adhere to the CG as his system reference point, also when returning to ship motions later. STEP 3: Measurement of volumes and volume centroids In stark contrast to BOUGUER, EULER confines himself to analytical definitions of stability criteria, volumes, centroids, areas, moments of inertia etc. He does not address their numerical evaluation at all. He is taking it for granted that once the shape of the ship or body is defined by some function the integrations can be readily performed. In his examples he usually deals with simple shapes where the integrations can be performed in closed form. STEP 4: Stability criterion In Chapter III, the main chapter on hydrostatic stability, EULER defines his stability criterion right away (Proposition 19) by: “The stability which a body floating in water in an equilibrium position maintains, shall be assessed by the restoring moment if the body is inclined from equilibrium by an infinitesimally small angle”. EULER illustrates this principle by discussing cases of unstable, neutral and stable equilibrium of ships and adds that it is necessary to quantify stability in terms of the restoring moment because even a stable ship may be in danger by external heeling moments and may require righting moments of greater magnitude. (The issue is not only whether the ship is initially stable or not, but also how much “stability capacity” it has). STEP 5: Evaluation of the criterion EULER then enters into the determination of the restoring moments by first examining a planar cross section of arbitrary shape (or thin disk) floating upright (Fig. 6). The Figure AMFNB = “AFB” is inclined by a small angle dw so that the new floating condition has the waterline ab. Since the center of gravity G remains his reference point, also for later purposes, he draws the parallel lines MN and mn to the waterlines before and after inclination, also through G. The center of buoyancy of the cross section is designated by O. A normal VOo to the inclined waterline is drawn through O. The equality of the immersed and emerging wedges requires that ab intersect AB at the center point C so that AC = BC.

166 H. Nowacki and L.D. Ferreiro

Fig. 6 EULER’s figure for centroid shift in inclined cross section, 2D case, (Euler 1967, fig. 39).

Let the displacement per unit length and the equal buoyancy force of the cross section before inclination be denoted by M = γ (AFB). For the inclined position the restoring moment is composed of three contributions: 1. The effect of the original submerged volume forming a positive restoring couple of forces through G and V: M GV = M GO dw

(5)

2. The effect of the submerged wedge CBb whose cross section area is:

BC 2 dw AB 2 dw = 2 8

(6)

and whose restoring moment about G hence is:

AB 2 dw γ (qo + GV) 2 where

qo =

(7)

2 1 Cb = AB 3 3

3. Likewise for the emerging wedge ACa the restoring moment is

AB 2 dw (po - GV) -γ 8 where

po =

(8)

2 1 Ca = AB 3 3

For all moments combined, replacing γ by M/ (AFB):

Historical Roots of the Theory of Hydrostatic Stability of Ships 167

M (AB 2 dw )(po  qo) M REST = M GO dw + 8AFB AB 3 ] Mdw [GO + 12AFB

=

(9)

The expression in square brackets has the dimension of a length and is the now well known result, corresponding to, in BOUGUER’s later terminology, the metacentric height of the cross section. This is how EULER’s restoring moment and BOUGUER’s metacentric height are connected. Note that the GO term reverses sign if G lies above the center of buoyancy O, as is common in cargo ships. EULER discusses this result at some length and for several simple shapes. In Proposition 29 EULER then arrives at the general three-dimensional case of a floating body of arbitrary, in particular asymmetrical shape, whose waterplane is drawn in Fig. 7. He denotes: M = displacement or weight of body V = submerged volume GO = distance between CG and CB, positive for G above B CD = reference axis of waterplane through centroid of waterplane area, parallel to axis through G CX = x = abscissa from origin C XY = y = ordinate in upper part of waterplane XZ = z = ordinate in lower part of waterplane p and q = area centroids of upper and lower parts of waterplane P and Q = equivalent pendulum lengths of waterplane area parts w.r.t. axis CD With

 y dx 2  ydx 2

pr =

3 2  y dx PR = 3 y 2 dx 

 z dx qs = 2  zdx 2

3 2  z dx QS = 3 z 2 dx 

168 H. Nowacki and L.D. Ferreiro

Fig. 7 EULER’s figure for the derivation of the stability criterion for a body of arbitrary shape, 3D case (Euler 1967, fig. 48).

EULER derives, since the axis CD runs through the waterplane centroid: ∫ y2 dx = ∫ z2 dx, and hence

 ( y  z )dx PR + QS = 3  y dx 3

2

3

2

(10)

from which in analogy to the planar case the restoring moment, divided by the angle dw, for the ship becomes:

M (GO +

 (y

3

 z 3 )dx 3V

)

(11)

In the special case of port/starboard symmetry (y = z), with IT =

I 2 3 ∫ y dx; and T = OM V 3

In our familiar notation (GO= GB , OM= BM ) the restoring moment simplifies into: M ( GB + BM ) = M GM

(12)

Historical Roots of the Theory of Hydrostatic Stability of Ships 169

STEP 6: Results This summarizes the course of steps EULER took to derive the initial restoring moment. EULER never used the word metacenter. EULER’s result thus is the initial restoring moment, divided by (M dw), which he uses as his stability criterion. Again EULER illustrates this result by many examples for simple shapes of solids, even by an analytical formulation for a shiplike body with parabolic section shapes. But he does not present any numerical calculations for an actual ship although he discusses many practical implications of his results. 3.2.4 Implications of the Stability Criterion

EULER interprets his results for practical applications to ships, mainly in Part II of “Scientia Navalis”. For EULER ships are “floating objects, carrying a cargo or payload, symmetrical starboard and port, propelled by rowing and/or sails”. Among the major conclusions he draws, we quote: Stability is judged by the restoring moment divided by the angle dw. Transverse and longitudinal stability are clearly distinguished and both addressed. EULER also derives an expression for combined heel and trim under oblique sail force moments, deriving the oblique restoring effects by rotation of the reference axis from the principal axes. To assess stability you need to know the ship’s displacement, the centroids CB and CG, the waterplane area and shape, which infers the restoring moment (without using the metacenter). To improve stability, lower the CG, raise the CB and/or widen the beam. The addition, removal and internal shifts of weight as well as the role of ballast for stability are discussed. The use of doubling (“soufflage”) with its pros and cons is mentioned. In Chapter IV of Part 1 EULER addresses practical problems of inclining ships under external moments or by internal weight shifts, though again only analytically, not numerically. He also addresses the issue of external wind loads and required stability, especially for small ships vs. big ships, cargo ships vs. war ships. EULER finally also makes the interesting “philosophical” remark that sailors knew about measures of stability all along. When the sailors say, “This ship can sustain a strong wind in its sails”, then EULER claims they mean the same thing as he expresses by his restoring moments.

3.3 Comparison of Approaches Based on the facts presented above it is now possible and in fact intriguing to compare the approaches taken by BOUGUER and EULER in their creative work on hydrostatic stability. This will bring out certain commonalities, but also underscore the differences. We have examined several aspects.

170 H. Nowacki and L.D. Ferreiro

3.3.1 Chronology:

BOUGUER worked on the subject of ship hydrostatic stability from about 1732 to at least 1740, and his treatise was published in 1746. EULER was engaged with this topic between about 1735 to at least 1740, “Scientia Navalis” appeared in 1749. During these formative periods of their new concepts there existed no contacts and no communications between them. Thus their work originated independently. But after their first scientific acquaintance through their participation in the 1727 Paris Academy award competition it is likely that they both underwent a common fermentation period on related subjects. Probably they were both looking for the missing pieces in the mosaic of sailing ship maneuvering, among which the hydrostatic response was foremost. It appears that their deeper familiarity with ARCHIMEDES originated during the period between 1727 and 1735. 3.3.2 Sources:

BOUGUER writes that he was familiar with the work of PARDIES, HOSTE, RENAU, most likely also with relevant publications by HUYGENS and the BERNOULLIs. He shows close acquaintance with ARCHIMEDES’ principles, although he does not mention his name. On calculus he quotes the well-known treatise by NEWTON and Roger COTES, but very likely he also thoroughly read de l’HÔPITAL, whose notation in the LEIBNIZ style he preferably uses. His knowledge of calculus was mostly self-taught, based on occasional tutorials and recommendations from mathematics professor Charles René REYNEAU, who wrote the basic textbook on the subject, “Analyse démontrée” in 1708. EULER also knew ARCHIMEDES, HOSTE, RENAU, whom he mentions. He learnt calculus first-hand from the BERNOULLIs, who stood firmly in the LEIBNIZ tradition. EULER’s book on mechanics, “Mechanica” in 2 volumes, where he displays exquisite knowledge of the principles of equilibrium and motions of mechanical systems, had appeared in 1736. Although both authors are not very generous in giving references on their sources it is possible to a certain extent to trace their intellectual ancestry by looking at their scientific and technical terminologies, even if they wrote in different languages, French and Latin. We have analyzed their vocabularies in their main treatises on hydrostatic stability and identified commonalities and distinct differences. They shared much common ground by using such established words as equilibrium, stability, weight, buoyancy, specific weight, centers of gravity and buoyancy, inclination (derived from ARCHIMEDES) and pressure, hydrostatics (from STEVIN and PASCAL). BOUGUER is unique in inventing the metacenter and using the evolute (from HUYGENS), but never the word “restoring moment”, while EULER’s vocabulary was vice versa. This indicates where their approaches differed.

Historical Roots of the Theory of Hydrostatic Stability of Ships 171

3.3.3 Stability Criteria:

Sections 3.1 and 3.2 have shown that BOUGUER and EULER derived equivalent stability criteria, but used different approaches. To sum up the comparison: Premises and axioms: BOUGUER’s work is founded on the premises of ARCHIMEDES, though augmented by a hydrostatic pressure law (STEVIN). EULER defines the axioms of hydrostatics a bit more strictly by postulating the direction independence of pressure and its normality to a surface. Magnitude of buoyancy force: Both agree and reconfirm the law of ARCHIMEDES by pressure integration and calculus. Measurement of volumes and centroids: BOUGUER takes the route of establishing numerical procedures, based on quadrature rules, for measuring volumes, centroids, areas etc. before giving analytical definitions of the stability criterion. In passing he skillfully adapts the “method of trapezoids” to shipbuilding applications. EULER skips this step entirely and drives directly at analytical formulations for his stability criterion. Stability criterion: BOUGUER’s invention and choice for a measure of stability is “metacentric height”, a geometric quantity representative of initial stability capacity. EULER chooses the “initial restoring moment”, a physical quantity, also measuring the ship’s initial stability capacity. Although the two ideas are closely connected, their meanings are conceptually different. BOUGUER never refers to “restoring moments”, EULER never uses “metacenter”. Neither author recurs to ARCHIMEDES’ idea of a “righting arm”. Evaluation of the criterion: BOUGUER derives the metacenter mainly from geometric arguments and, allowing for the shift of wedge volumes and hence CB, determines the metacenter as the point of intersection of the upright and inclined buoyancy directions. EULER uses physical arguments of moment equilibrium and shift of volumes and finds the restoring moment simply by a summation of moments. Results: BOUGUER’s result is metacentric height, a distance. EULER’s result is a restoring moment. The two results are equivalent in practice, but not equal in approach.

3.3.4 Audience, Style and Language:

BOUGUER’s “Traité du navire”, the first modern synthesis of theoretical naval architecture, is written in French for a readership of scientists and constructors who are to be introduced to the use of theory in practical ship design and shipbuilding. BOUGUER benefited from both his experience as a hydrographer and his collaboration with ship constructors. His style is clear and logical, explaining many practical details, almost as in a textbook. He is concerned about how this methodology can be implemented. This style does not cause any sacrifices in rigor. His language is lucid, his own new ideas come across crisply.

172 H. Nowacki and L.D. Ferreiro

EULER in his Latin text of “Scientia Navalis” writes as an applied mechanicist/ physicist and addresses mainly the scientific community of his era. Among those who were educated to read scientific treatises of this sort in Latin, which was still a lingua franca in the academic world, EULER’s brilliance of style, his inimitable logic and clarity were highly praised. These attributes also apply to his “Scientia Navalis”. His work on hydrostatic stability is still valid today. But at the same time he did not have any experience with practical shipbuilding applications. Thus he left many details for implementation by later successors, which delayed the spreading of his methods, certainly in comparison to BOUGUER. Yet in the final balance the scientific and engineering community must be grateful to both BOUGUER and EULER.

3.4 Synopsis Table 1 gives a synopsis of important milestones and ideas in the development of the theory of hydrostatic stability by indicating where certain elements of this theoretical knowledge first occurred. The importance of ARCHIMEDES ’s physical insights and the relevance of BOUGUER’s and EULER’s parallel discoveries described by calculus become clearly visible in this condensed tabular summary of historical steps. Table 1. Historical development of stability concepts

Historical Roots of the Theory of Hydrostatic Stability of Ships 173

4 Further Work 4.1 Practical Applications Stability theory quickly found direct applications in the day-to-day practice of ship design. This occurred in two ways: first, by the increasing sophistication of calculations for weights, centers of gravity and the metacenter within the design process; and second, by the use of inclining experiments to validate stability. These applications were surprisingly widespread; stability theory was quickly incorporated in navies where there was already a strong institutional development of scientific naval architecture, notably in France, Denmark, Sweden and Spain. Those countries also created schools of naval architecture during the 18 th century, where students were weaned on stability theory and naturally used it when they became constructors at the dockyards. This was not the case for the British or Dutch navies, which had provided little direct support for scientists working on ship theory, and which did not establish permanent schools of naval architecture until the 19th century. 4.1.1 Stability Calculations During Design

The most famous of these schools of naval architecture was the Ecole du Génie Maritime, established in 1741 in France by the scientist Henri Louis DUHAMEL DU MONCEAU (1700-1782), whom MAUREPAS had appointed to InspectorGeneral of the Navy. DUHAMEL worked with BOUGUER to create the first comprehensive textbook for the students, based on “Traité du navire”. DUHAMEL’s genius was to take BOUGUER’s complex mathematics and render them into step-by-step instructions on how to calculate the metacenter. The textbook, “Elémens de l’architecture navale” (Duhamel 1752) became the standard reference for both students and constructors. DUHAMEL DU MONCEAU was also involved in developing the Ordinance of 1765 under Navy minister Etienne François, Duke of CHOISEUL, which created the Corps du Génie Maritime and formalized the data to be included on ship’s plans: “centers of gravity and resistance, and height of the metacenter” as well as accompanying calculations and a tabulation of hull materials. Later, the standardized plans introduced in 1786 by DUHAMEL’s successor as InspectorGeneral, Jean-Charles de BORDA, listed the specific immersion of the hull at full and light load in tonnes-per-cm equivalent, which allowed a rapid estimate of the effect of loading weights on the ship.

174 H. Nowacki and L.D. Ferreiro

4.1.2 Inclining Experiments

Another facet of rapid adoption of the metacenter was that the first recorded inclining took place in 1748, just two years after BOUGUER published his work. FrancoisGuillaume CLAIRIN-DESLAURIERS, then a junior constructor at Brest, performed the experiment on the newly-built 74-gun ship Intrépide, apparently out of curiosity to test the new theory. He hung two 24-pound cannons (each weighing over 2 tonnes) from a buttress built on the side of the ship (Fig. 8), and although the buttress broke, he was able to take enough measurements to ascertain that the ship’s GM was 1.8m (Clairin-Deslauriers 1748).

Fig. 8 CLAIRIN-DESLAURIERS’ drawing of the inclining of Intrépide (Clairin-Deslauriers 1748)

4.1.3 CHAPMAN and SIMPSON’s Rule

Another important contribution to the promulgation of hydrostatic stability calculations came from the Swedish naval constructor and innovative ship designer Fredrik Henrik CHAPMAN. CHAPMAN (1721-1808), the son of the superintendent of the Göteborg dockyard, was endowed by this background with much practical shipbuilding knowledge. He visited Britain in 1750 to improve his knowledge in mathematics and other fundamentals. He took classes from Thomas SIMPSON (1710-1761), a gifted mathematician trained in NEWTON’s tradition and an instructor at the Royal Military Academy in Woolwich near London. SIMPSON reportedly taught CHAPMAN a quadrature rule for planar curves, which yields a numerical approximation of the integral of a continuous function using at least one equally spaced double interval, hence at least three offset points per double interval, or any desired number of further double intervals of the same spacing. This quadrature rule later became known as “SIMPSON’s Rule”, especially in shipbuilding, although the rule has several historical precursors and SIMPSON never claimed to have invented it. CHAPMAN, after further visits to France and Holland and having picked up much valuable background including

Historical Roots of the Theory of Hydrostatic Stability of Ships 175

the recent findings by BOUGUER on the metacenter, returned in 1757 to Sweden to continue his career as an important ship constructor. In this process he soon made it his habit to calculate ship displacements (1767), also as a function of draft, and assessing the metacenter, applying SIMPSON’s Rule for numerical integration. His engineering methods applied in ship design became well known internationally soon after the publication of his book Tractat om Skepps-Byggeriet (Chapman 1775), where he explained his calculations in much detail (Harris 1989). Thus CHAPMAN helped to promulgate the use of the metacenter and to make its calculation a routine matter among naval architects. At the same time SIMPSON’s Rule, which offers certain efficiency advantages over the trapezoidal rule for an equal number of intervals, was popularized and is still much favored today.

4.2 Extensions of Stability Theory 4.2.1 Hydrostatic Stability About Other Axes

In their work on the masting of sailing ships in 1727 it must have occurred to both BOUGUER and EULER that in general the external wind moments would act in an oblique direction to the ship’s centerplane so that heeling and trimming would result simultaneously. EULER remembered this open question and in “Scientia Navalis”, Part 2, returned to this issue. He derived an expression for the restoring moment about an oblique axis between the axes of heel and trim. The moments of inertia about this axis could be constructed, when the moments of inertia IT and IL for transverse and longitudinal stability were known, by using an ellipse of inertia moment construction. Thus this method was able to predict at least the initial hydrostatic response (small angles) to a wind moment in an oblique plane in terms of simultaneous heel and trim angles. EULER mentioned on the side, too, that longitudinal stability was always much greater than transverse stability. It was some time later that the Spanish naval constructor and engineering scientist Jorge JUAN Y SANTACILIA (1713-1773), who knew BOUGUER and his work very well from the Peruvian expedition and also was in correspondence with several European Academies, concerned himself not only with ship stability, but also with ship oscillations, especially with pitching motions. In this context he extended BOUGUER’s concepts of the metacenter to longitudinal inclinations and first introduced the definition of the longitudinal metacentric height GML. He pointed out that it by far exceeds the transverse metacentric height. His book “Examen Maritimo”, which appeared in 1771, combined the theoretical insights of his days with much practical design experience and became a widely used reference and textbook (Juan y Santacilia 1771).

176 H. Nowacki and L.D. Ferreiro

4.2.2 Large Angles of Heel

A broader extension of stability theory occurred not on the European continent but in Britain. In two papers presented to the Royal Society of London, the British mathematician George ATWOOD examined the inclination of ships at large angles of heel. The first paper (Atwood 1796) based on a thorough knowledge of ARCHIMEDES, BOUGUER and EULER, reviews the fundamental physical principles of hydrostatic stability, applied to finite angles of heel. Here ATWOOD investigates the stability properties of homogeneous solids of simple shape (parallelepiped, cylinder, paraboloid) through 360 degrees of rotation as a function of body draft at rest (or specific weight). He finds numerous equilibrium positions (8 or 16), only some of them stable. He develops a shifted wedge volume method for finite heeling angles and reviews the numerical quadrature rules, settling for STIRLING’s 3 interval rule. This paper already drew full attention to the fact that stability must be judged over a range of finite, practical heel angles; initial stability alone is inadequate as a stability measure. In the second paper (Atwood and Vial du Clairbois 1798), which ATWOOD coauthored with the French constructor Honoré-Sébastien VIAL DU CLAIRBOIS, the investigation was extended to actual ships, and for the first time a numerical analysis of the “righting moments” of ships over a large range of heeling angles was performed. ATWOOD and VIAL DU CLAIRBOIS introduced the term GZ for the “righting arm”, which was again numerically evaluated by a wedge volume shift method. This successfully demonstrated the feasibility of numerical stability analysis for actual ships over a range of finite heeling angles. The necessity of performing such calculations rather than just relying on initial stability measures was again underscored. Yet in shipbuilding practice it still took several more decades before the initial difficulties in numerical integration could be overcome by more robust methods and instruments. Even half a century later such stability evaluation had not yet become a routine matter. It was not before the early 19th century that the knowledge on the metacenter progressed one further step by the work of French constructor and mathematician Charles DUPIN from metacentric curves to metacentric surfaces (Dupin 1813). DUPIN stated that the ship for finite angles of inclination in whatever direction possesses a surface of centers of buoyancy. This surface at every point has two principal curvatures, thus one can construct two sheets of metacentric surfaces, i.e., evolute surfaces of the buoyancy surface. The two metacenters for the upright condition, MT and ML, each is a special point on one of the surfaces. It is probably fair to say that at this level of perception the issue of hydrostatic stability had been fully resolved in terms of the relationships between ship geometry, physical responses to inclinations, and differential geometry of metacentric surfaces for finite angles of heel.

Historical Roots of the Theory of Hydrostatic Stability of Ships 177

5 Conclusions ARCHIMEDES laid the foundations for the stability of floating systems, introduced a stability measure similar to the righting arm and presented an approach for assessing the ability of a floating inclined solid to right itself. But his applications were limited to simple geometrical shapes. Fortunately his manuscript “On Floating Bodies” survived in a few copies and became more accessible again by a Latin translation in the 13th c. and also in print after 1500. Yet it took a few more centuries before modern hydrostatic stability was established and could be applied to actual ships, in particular also at the design stage. The interest in scientific solutions for ship stability gained new momentum for practical and scientific reasons by about 1700: The leading navies were getting more concerned about stability risks with increasing ship sizes, gunports that were close to the water and increasingly heavier armament. In the beginnings of the Age of Enlightenment, new expectations were raised with regard to the capabilities of science to predict physical phenomena and the performance of technical systems. Mathematical breakthroughs occurred in infinitesimal calculus, analytical and numerical methods. The modern age of engineering science made rapid progress in mechanics and hydromechanics, including the study of equilibrium and stability of mechanical systems. All of this contributed to an atmosphere in the scientific community in the early 18th c. that was open to new challenges, also in application to ships. Both BOUGUER and EULER were actively involved in these general developments and certainly exposed to this unique zeitgeist. BOUGUER responded to the recognized problem of hydrostatic stability more as an engineering scientist, EULER reacted rather more like an applied mechanicist and mathematician. Both were able to reformulate and solve this problem in their own unique and original ways. As our comparisons in this paper have reconfirmed, BOUGUER’s and EULER’s nearly simultaneous work was not only performed quite independently, which was never doubted, but was also distinctly different in approach and justification. Both investigated the ability of the ship to right itself after an infinitesimal heeling displacement. BOUGUER reasoned mainly geometrically and for the inclined ship derived a length measure, the metacentric height, as the decisive geometric measure of initial stability. EULER argued mainly on mechanical grounds and deduced the initial restoring moment as his measure of hydrostatic stability. Despite these contrasting styles of justification, as we appreciate today of course, both stability measures are in fact equivalent and can be converted into each other. We may still benefit from both lines of thought and should be grateful for the insights we owe to these two congenial pioneers.

178 H. Nowacki and L.D. Ferreiro

We have explained why it took several more decades before these landmark discoveries in ship stability were fully accepted and widely applied in shipbuilding practice. But the foundations laid by BOUGUER and EULER have remained of lasting value as secure cornerstones in our knowledge for designing safe and stable ships.

Acknowledgments The authors are pleased to acknowledge the support and encouragement they received during the preparation of this work from many colleagues, in particular from Jean DHOMBRES, Emil A. FELLMANN, Jobst LESSENICH and Gleb MIKHAILOV. The first author also expresses his gratitude to the Max Planck Institute for the History of Science Berlin, where he is a Visiting Scientist and where his work was much promoted by personal communications and the excellent library resources.

References Archimedes (2002) The Works of Archimedes, edited and translated by T.L. Heath, repub by Dover Publ, Mineola, N.Y. Atwood G (1796) The Construction and Analysis of Geometrical Propositions, Determining the Positions Assumed by Homogeneal Bodies which Float Freely, and at Rest, on a Fluid’s Surface; Also Determining the Stab of Ships and of Other Float Bodies, Philos. Trans. of the R. Soc. of London, Vol. 86: 46-278. Atwood G, Vial du Clairbois HS (1798) A Disquisition on the Stability of Ships, Philos. Trans. of the R. Soc. of London, Vol. 88, vi-310 Bouguer P (1727) De la mâture des vaisseaux, pièce qui a remporté le prix de l’Acad. R. des Sci. de 1727, (On the masting of ships) Jombert, Paris. Bouguer P (1746) Traité du Navire, de sa Construction et de ses Mouvemens (Treatise of the Ship, its Construction and its Movements), Jombert, Paris Boyer CB (1949) The History of Calculus and its Conceptual Development, 1939, reprint Dover Publ, N.Y. Burckhardt JJ, Fellmann EA, Habicht W, eds (1983) Leonhard Euler 1707-1783, Mem. Monogr., issued by the Canton Basel-City. Birkhäuser Verlag, Basel. Calinger R, Leonhard Euler (1996) The First St. Petersburg Years (1727-1741), Hist. Math., vol. 23, 121-166, Academic Press. Cerulus FA (2001-Jan) Un mathématicien mesure les navires (A mathematician measures the ships), in Géomètre; édition des Histoires de la mesure, Publi-Topex editions, Paris, 34-37. Chapman FH af (1968) Tractat om Skeppsbyggeriet (Treatise on Shipbuilding), Johan Pfeffer, Stockholm (1775); translation and facsimile with commentary in Architectura Navalis Mercatoria (Merchant Nav Archit), Praeger Publ, N.Y. Clairin-Deslauriers G (1748-May 8) Letter to Duhamel du Monceau, Peabody-Essex Museum, collection Duhamel du Monceau, MH 25 box 2 folder 2 item 4.8; Bres. Czwallina-Allenstein A (ed.) (1996) Treatises of Archimedes, translated and annotated by the editor in 1922, republished in Ostwald’s Klassiker der exakten Wissenschaften, vol. 201, Verlag Harri Deutsch, Frankfurt am Main.

Historical Roots of the Theory of Hydrostatic Stability of Ships 179 Deane Sir A (1981) Deane’s Doctrine of Naval Architecture (1670), edited by Brian Lavery, Conway Maritime Press, London, 71-73. Demerliac A (1995) La Marine de Louis XV: Nomenclature des navires français de 1715 à 1774, (The Navy of Louis XV: nomenclature of French ships from 1715 to 1774), Editions Omega, Nice, p. 68. Dhombres J (1999) Mettre la géométrie en Crédit: Découverte, Signification et Utilisation du Métacentre Inventé par Pierre Bouguer (Giving Credit to Geometry: Discovery, Significance and Use of the Metacenter invented by Pierre Bouguer), in Sci. et Tech. en Perspect.”, série 3, fasc. 2, 305-363, Paris. Dijksterhuis EJ (1987) Archimedes, translated from Dutch by C. Dikshoorn, Princeton Univ Press, Princeton, N.J. Duhamel du Monceau HL (1752) Elémens de l’architecture navale (Elements of naval archit), Jombert, Paris. Dupin C (1813) Développements de géometrie: avec des applications à la stabilité des vaisseaux, aux déblais et remblais, au défilement, à l’optique, etc. pour faire suite a la Géometrie descriptive et a la Géometrie analytique de M. Monge (Developments in geometry; with applications to the Stab of vessels, to trenches and embankments, to ramparts, to optics, etc. following the Descriptive Geometry and Analytic Geometry of M. Monge), Mme Ve Courcier, Paris (1813). Euler, L. Leonhard Euler. Letters on Scientific Subjects, letters R 2701 (20 June 1737) and R 2705 (12 April 1740), from Euler to Wegersløff, in German with Russian translation, ed by the Acad. of Sci. of the USSR, Moscow-Leningrad (1963). Euler L. Scientia Navalis seu Tractatus de Construendis ac Dirigendis Navibus (Science of Ships or Treatise on How to Build and Operate Ships), 2 vols., St. Petersburg, (1749). Republ in Euler’s Collected Works, Series II, vols. 18 and 19, issued by the Euler Comm of the Swiss Acad of Natural Sci, Zurich and Basel (1967 and 1972). Euler L (1972) Judicium de Libello Domini de LA CROIX ex Gallia Huc Transmisso (Review of the Booklet by M. de LA CROIX from France Submitted Here) (1735); and: Notae ad Responsiones Viri Illustris de LA CROIX Factae ab L. EULER (Notes on the Response by the Famous M. de LA CROIX Made by L. EULER) (1736), Appendices I and II in vol. 19, Series II, Euler’s Collected Works, issued by the Euler Comm of the Swiss Acad of Natural Sci, Basel. Euler L (1974) Meditationes super Problemate Nautico, quod Illustrissima Regia Parisiensis Academia Scientiarum Proposuit (Thoughts on the Nautical Problem Proposed by the Most Illustrious Paris Royal Acad of Sci, vol. 20, Series II, Euler’s Collected Works, issued by the Euler Comm of the Swiss Acad of Natural Sci, Commentationes Mech et Astronomicae, Basel. Fellmann EA: Leonhard Euler, in German, Rowohlts Monographien, volume 387, Rowohlt Taschenbuch Verlag, Reinbek (1995), translated by E. Gautschi and W. Gautschi, in English, Birkhäuser Verlag, Basel Boston Berlin (2007). Ferreiro LD (2007) Ships and Science: The Birth of Naval Architecture in the Scientific Revolution, 1600-1800, MIT Press, Camb. Habicht W (1974) Introduction to Vol. 20 of the 2nd Series, in German, pp. VII-LX in Euler’s Collected Works, Series II, vol. 20. Harris DG and FH Chapman (1989) The First Naval Architect and His Work, Conway Marit Press, London, 20, 215-220. Heiberg JL and Zeuthen HG (1906-07) A New Treatise by Archimedes, in German, Bibliotheca Mathe, B.G. Teubner Verlag, Leipzig. Hoste P (1697) Théorie de la construction des vaisseaux (Theory of the construction of vessels), Arisson & Posule, Lyon. Huygens C (1967a) De iis quae liquido supernatant libri tres (Three books on bodies floating on top of a liquid) (1650). Published in Œuvres Complètes de Christiaan Huygens, vol. XI (1908). Repub Swets and Zeitlinger, Amsterdam.

180 H. Nowacki and L.D. Ferreiro Huygens C (1967b) Horologium Oscillatorium…, (Pendulum Clockwork…), Part III, F. Muguet, Paris, 1673. In Œuvres Complètes de Christiaan Huygens, vol. 18 (1934). Swets and Zeitlinger, Amsterdam. Juan y Santacilia J (1771) Examen marítimo, Theórico Práctico (Maritime examination, theoretical and practical), Manuel de Mena, Madrid. La Croix CM de (1736) Eclaircissemens sur l’Extrait du méchanisme des mouvemens des corps flotans (Clarifications on the extract of the mech of the movements of floating bodies), Robustel, Paris. Netz R, Noel W (2007) The Archimedes Codex. Revealing the Secrets of the World’s Greatest Palimpsest, Weidenfeld & Nicholson, Great Britain. Mairan JJ d’Ortous de (1724) “Instruction abregée et méthode pour le jaugeage des navires” (Abridged instruction and method for admeasuring ships), in Hist. et Mem. de l’Acad. des Sci. 227-240 Mikhailov GK (1983) Leonhard Euler and the Development of Theoretical Hydraulics in the 2nd Quarter of the 18th Century, in German. 229-241, Burckhardt et al. Nowacki H (2002) Archimedes and Ship Stability, Proc. Intl. Multi-Conf./Euroconf. on Passenger Ship Design, Operation and Safety, Anissaras/Crete, edited by A. Papanikolaou and K. Spyrou, NTU Athens (Oct. 2001). Republ as Preprint No. 198, Max Planck Inst for the Hist of Sci, Berlin. Stevin S (1586) The Principal Works of Simon Stevin, 5 vols, ed. E.J. Dijksterhuis, C.V. Swets and Zeitlinger, Amsterdam (1955). In Vol I (General Introd, Mech): De Beghinselen des Waterwichts (Elements of Hydrostatics) and in the supplement to Beghinselen der Weegkonst (Elements of the Art of Weighing) the short note: Van de Vlietende Topswaerheit (On the Floating Top-Heaviness), first publ in Dutch. Truesdell C (1954) Rational Fluid Mechanics, 1687-1765, in Euler’s Collected Works, Series II, vol. 12, pp. VII-CXXV, issued by the Euler Comm of the Swiss Acad of Natural Sci, Lausanne. Truesdell C (1983) Euler’s Contributions to the Theory of Ships and Mechanics, an essay review, in Centaurus, vol 26, 323-335. Witsen N (1994) Aeloude en hedendaegsche scheeps-bouw en bestier (Ancient and modern shipbuilding and handling), Casparus Commelijn; Broer en Jan Appelaer Amsterdam (1671). Facsimile by Canaletto-Alphen Aan Den Rijn, Amsterdam (1979). Reprinted with commentary in Hoving, A.: Nicolaes Witsens Scheeps-Bouw-Konst open gestelt, (Nicolaes Witsen’s System of Ship Building), Uitgeverij Van Wijnen, Franeker.

The Effect of Coupled Heave/Heave Velocity or Sway/Sway Velocity Initial Conditions on Capsize Modeling Leigh S. McCue* and Armin W. Troesch** *A erospace and Ocean Engineering, Virginia Tech **Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, Michigan

Abstract

Obar et al. (2001) described a series of model tests and a numerical model of a rectangular barge in regular beam seas with three degrees of freedom in roll, heave, and sway used to investigate the onset of capsize. The model had minimal freeboard resulting in significant water-on-deck. This paper employs the numerical model to examine specifically the impact of the coupling in heave with heave velocity initial conditions or in sway with sway velocity initial conditions on ultimate stability. Direct comparison with experimental results was used to validate the numerical model and can be found in Lee et al (Lee, 2001; Lee et al, 2006).

1 Introduction One of the greatest threats to vessel safety is capsize. While capsize stability has been experimentally studied in great detail for one degree of freedom, see for example (Soliman and Thompson, 1991; Thompson, 1997), at the time of the 2003 Ship Stability conference, little had been done with multiple degree of freedom experiments in which great attention was paid to initial conditions. This work extends the experimentation conducted by Obar et al. (2001) and the numerical simulation and experimental structure developed by Lee et al. (Lee, 2001; Lee et al, 2006) to the case of coupled state variables. Vassalos and Spyrou (1990) demonstrated that combinations of effects (in their case the effect of directional instabilities on transverse stability) can dramatically alter the ultimate stability of a vehicle. Thompson and de Souza (1996) show the dependence of stability on nonlinear coupling between roll and heave. While Soliman and Thompson (1991) and Thompson (1997) as well as the work of Taylan (2003)

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_9, © Springer Science+Business Media B.V. 2011

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182 L.S. McCue and A.W. Troesch

thoroughly investigate the one degree of freedom problem, Vassalos and Spyrou (1990) and Thompson and de Souza (1996) make clear the importance of studying combined effects. Murashige and Aihara (1998) investigate the case of a forced flooded ship experimentally and mathematically. While they improve upon the traditional onedimensional model through inclusion of the effects of water on deck, their model is based upon assumptions that cause one to neglect sway and heave degrees of freedom. The work presented in this paper seeks to expand upon the one degree of freedom models previously discussed, and the computationally intensive models developed for fully nonlinear simulation and/or water on deck models which are currently being developed and validated (see for example Huang et al, 1999 and Belenky et al, 2002). This paper discusses results from a quasi-nonlinear time domain model developed by Lee (2001) which is far less computationally intensive than its fully nonlinear counterparts allowing for investigation of far greater quantities of data. Additionally, the numerical model is validated by three degree of freedom experimental results conducted at the University of Michigan Marine Hydrodynamics Laboratory (Obar et al, 2001; Lee, 2001, Lee et al, 2006). Over one hundred eighty separate experiments were used in the study to validate the numerical model (Obar et al, 2001; Lee, 2001, Lee et al, 2006). For each test case a simple box barge was excited at a frequency of 6.8 rad/s. The waves acting upon the model had a wave height to wave length ratio of 0.02. Each release, however, featured different initial conditions in roll, roll velocity, sway, sway velocity, heave, and heave velocity. These values were measured in 1/30th of a second increments for the duration of the experimental run. Approximately 37% of the test runs resulted in capsize suggesting that initial conditions were a significant contributor to subsequent dynamics. While roll initial conditions (i.e. initial roll angle and roll velocity) are well known to influence capsize (e.g. Thompson, 1997), little has been said about heave initial conditions. A reducedmodel nonlinear time-domain computer program is used in this work to provide guidance in interpreting the experimental results. Qualitative behavior of the capsize dynamics is presented in a number of formats. The roll phase space is characterized by “safe/unsafe basins” (e.g. Thompson, 1997). These figures show capsize boundaries based upon roll initial conditions. The evolution or change of these boundaries is investigated for varying sets of heave and sway initial conditions. From similar analyses, “integrity curves” are generated for systematic variations in the initial heave displacement and heave velocity as well as sway displacement and velocity. Each point on an “integrity curve” represents the ratio of safe (i.e. non-capsize) basin area for a given wave amplitude to a similar safe basin area derived from zero wave amplitude. In this way, the integrity curves show the relative influence of incident wave excitation on capsize relative to vessel safety in the absence of incident waves. Results from the simulation are presented here with discussions of accuracy and time of computation. Qualitative guidelines are suggested and recommendations are

Sway/Sway Velocity Initial Conditions on Capsize Modeling 183

made for determining whether experimental programs may be sensitive to test initial conditions.

2 Numeral Analysis 2.1 Definition of Numerical Model The works of Lee (2001) and Lee et al (2006) develop a quasi-nonlinear three degree of freedom ‘blended’ hydrodynamic model to simulate highly nonlinear roll motion of a box barge. The model uses an effective gravitational field to account for the centrifugal forces due to the circular water particle motion in addition to the earth’s gravitational field. This results in a local time dependent gravitational field normal to the local water surface. Additionally the model assumes long waves. Equation 1 below gives the equations of motion for the numerical model in global coordinates where a and b are added mass and damping coefficients, m and I are the mass and moment of inertia of the body, ge is the time dependent effective gravitation, f D are diffraction forces, is the time dependent displaced volume of water, and GZ is the ship’s time dependent roll righting arm.

m  a 22   0  a 42 

0 m  a33

b22 0  b42

b24   x g  0 0 0   x g    0   y g   0 0 0   y g b1  g  0 0 b2  g g 

0 b33 0

0

a 24   xg    0   yg   I cg  a 44  g      

(1)

 g e2   f 2D      g e3   mg  f 3D   D   g e4 GZ  f 4  The global and body fixed coordinate systems for the model are defined in Figure 1.

184 L.S. McCue and A.W. Troesch



Fig. 1 Coordinate system definition, Model scale definitions: 18.25 cm draft, 1.12 cm freeboard, 66 cm length, and 30.48 cm beam.

While the numerical model cannot account for the dynamics of water on deck, such as sloshing, it does account for hydrostatic effects due to deck immergence. The forcing due to the waves is modeled as a cosine wave. To simulate laboratory transients, the wave “ramps” from zero until steady state. An experimentally determined envelope curve was generated to create a ramped cosine wave that closely resembles the waves generated in the tank (Obar et al, 2001). All release times for the model are defined relative to the maximum transient wave crest as shown in Figure 2 below and indicated by the notation “to”.

Fig. 2 Sample wave profile (top) with enlarged region showing definition of points relative to to (bottom).

Sway/Sway Velocity Initial Conditions on Capsize Modeling 185

While this ‘blended’ model has admitted weaknesses due to the simplifications employed, it has the distinct advantage of computational efficiency. Other, multidegree of freedom models that simulate various dynamics of water on deck run significantly slower than real time (Belenky et al, 2002). However, this model, whose results are qualitatively experimentally validated (Lee et al, 2006), runs in a fraction of the actual run time. The work that makes up this numerical study represents years worth of real time data; using this quasi-nonlinear model such data can be collected in a matter of days.

2.2 Numerical results As discussed in the work of Soliman and Thompson (1991), one can consider a set of model releases in which all initial conditions, save roll and roll velocity, are held constant. A sweep is made of roll and roll velocity initial condition pairs with conditions resulting in capsize marked in black and non capsize left white yielding a ‘safe basin’. Figure 3 shows a sample safe basin generated with this numerical model (note e denotes excitation frequency, n represents vessel natural frequency in roll, similarly te and tn represent excitation and natural periods respectively).

Fig. 3 Safe basin for release time of to-2 with sway, sway velocity, heave, and heave velocity initial conditions set equal to zero at release. en≈3, h/=0.019.

186 L.S. McCue and A.W. Troesch

Fig. 4 Safe basin for release times starting from to-2 in 30 degree increments with en≈3, h/=0.019, roll angle from -30 to 30 deg and roll velocity from -30 to 30 deg/s; i.e. the left hand column represents safe basins at to-2, to-1.5, to-1, to-.5, to, to+.5, to+1, and to+1.5. All heave and sway displacements and velocities initially set equal to zero. x and y axes represent roll and roll velocity respectively.

Sway/Sway Velocity Initial Conditions on Capsize Modeling 187

One can then consider a set of safe basins with some third parameter varied. For example, Figure 4 shows a series of safe basins for different release times. The figure shows safe basins for 48 different release times starting from to-2 until to+2. Thus a shift between two safe basins represents a 30 degree shift in phase location along the wave profile (i.e. 1/12 ). This figure illustrates how crucial accurate accounting of time, and implicitly sway location, is on predicting vessel stability. Additionally, this plot demonstrates the impact transients in the incident wave profile have upon the safe basins. The safe basins before steady state, such as from to-2 to to-1, exhibit no periodicity (i.e. to-2 and to-1’s safe basins are quite different), while basins from to to to+2 produce a periodicity in that every 360 degrees of phase (or one wave length) the safe basin is virtually identical (i.e. to+.5 and to+1.5 are nearly the same). It should be noted that a change in sway displacement can be explicitly written as a shift in time. For example, upon reaching steady state, a sway displacement equal to one wave length at time tx could instead correspond to a sway displacement of zero at a time tx+1, that is time moved forward through one wave period. Thus a shift in time corresponds to a change in sway displacement if time is referred to by a particular location in the wave profile, e.g. the maximum wave crest, to (Figure 2). As described by Thompson (1997), one can then introduce the idea of integrity values. Integrity is a ratio of safe area for given initial conditions to safe area for some reference case. For a typical integrity curve, wave frequency and all non-roll initial conditions would be held constant while wave amplitude is varied. As an example, Figure 5 presents an integrity curve for the box barge numerical model described in the previous section. In Figure 5 sway, sway velocity, heave, and heave velocity are all set to zero at the instant of release. A safe basin is generated for varying roll and roll velocity initial conditions (between -20 to 20 degrees and –15 to 15 degrees/sec respectively). This is done for multiple wave amplitudes with integrity values being calculated as the ratio of safe area at a particular wave height to safe area for a wave height of zero. Thus the curve begins at one. The resulting curve is an integrity curve as defined by Thompson (1997) in which one notes the crucial characteristic ‘Dover cliff ’ (1997) at which point there is a distinct, rapid loss of stability. One can note that in Figure 5, integrity values are greater than one for wave amplitude to wave length ratios of 0 to approximately 0.007. Physically, integrity values greater than 1.0 indicate that the presence of waves at certain wave amplitudes can have a stabilizing effect upon the vessel relative to the zero wave condition.

188 L.S. McCue and A.W. Troesch

1.2

1

Integrity

0.8

0.6

0.4

0.2

0 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

Wave Amplitude per Wave Length (  o/ )

Fig. 5 Integrity Curve. Sway, sway velocity, heave, and heave velocity 0 at time of release. Te/Tn~3.

In Lee et al (2006) a detailed study is conducted considering the ways in which these curves shift for varying initial conditions of sway, sway velocity, heave, and heave velocity. This results in series of curves such as those found in Figure 6. In this case one sees the variations in the location of the cliff; however, direct comparison is difficult as the normalization factor for each curve varies. This is analysed in detail for the 6 state variables in Lee et al (2006). Heave (normalized by beam) 1.4

1.2

-0.09 -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

-0.09

1

Integrity

0.8

0.6

0.4

0.2 0.09

0 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

Wave Amplitude per Wave Length (  / )

Fig. 6 Heave Integrity Curves. Sway, sway velocity, and heave velocity 0 at time of release. Te/Tn~3. Note significant change with respect to heave in location of cliff on curve but not in xaxis intercept (location where integrity is zero).

Sway/Sway Velocity Initial Conditions on Capsize Modeling 189

While this lends great insight into anticipated behaviour from individual state variable variations, it still cannot fully capture the idiosyncrasies of experiments as there is no graphic coupling amongst the remaining state variables, sway, sway velocity, heave, and heave velocity. Therefore, instead of viewing the parameters as integrity curves, one can consider a surface, in which integrity (defined on the roll and roll velocity phase space plane) for a fixed wave amplitude is calculated for pairings of two other state variables such as sway and sway velocity (Figure 7) or heave and heave velocity (Figure 8). Each of these surfaces is for a fixed wave height and frequency (h/=.019 and en≈3) and is normalized such that the integrity value is 1 when sway, sway velocity, heave, and heave velocity are zero. Figure 7 presents a three dimensional integrity surface where each point is an integrity value representing the ratio of safe area to unsafe area for a roll/roll velocity grid as a function of sway and sway velocity. Heave and heave velocity are defined as zero at the time of release. Additionally excitation frequency and incident wave amplitude are constants. This figure clearly shows the relative independence of stability on sway velocity initial conditions. If one were to pick a fixed value of sway the plot collapses to a two-dimensional curve representing integrity as a function of sway velocity. This curve is relatively horizontal and constant thus showing the independence of integrity on sway velocity. For example, for sway equal to one wave length (or correspondingly zero wave length), the maximum integrity value as a function of sway velocity is 1.24 and the minimum is 0.70. While this is non-trivial, comparatively speaking, this is a relatively small change in contrast to the effects seen by changes in sway displacement, heave displacement, and heave velocity. If one were to instead consider a constant sway velocity, we see integrity values as a function of sway vary from zero to 2.2. Therefore one can see that for any fixed sway velocity value, there is a strong dependence of the integrity value upon sway position. In fact, for values of sway from approximately .15 to .65the integrity values are zero irregardless of sway velocity. This indicates for some initial release positions on a wave (the region 15%-65% of a wave length beyond a peak) the vessel shall always capsize regardless of initial roll angle, initial roll velocity, or sway velocity for a wave amplitude to wave length ratio of 0.0096 and an excitation frequency to natural frequency ratio of approximately 3. This is due to the fact that this wave amplitude is in a particularly steep region of the integrity curve. Referring to Figure 5, one should note the steep drop off at approximately o/=.01. For a set of sway integrity curves the location of this ‘cliff ’ is relatively invariant. More details on this are given by Lee et al (2006). Figure 8 demonstrates the intricate interdependence between heave and heave velocity state variables in determining ultimate vessel stability. For the surface presented in Figure 8 sway and sway velocity are zero at the time of release and

190 L.S. McCue and A.W. Troesch

wave height and frequency are constants for each point determining the surface. It should be noted though that after the time of release the vessel is unconstrained in all three degrees of freedom, thus sway and sway velocity are allowed to evolve in time just as heave, heave velocity, roll and roll velocity evolve. This is true for the data comprising both Figures 7 and 8. Unlike Figure 7 we note there are no similar statements we can make regarding a relative independence. Both heave and heave velocity have substantial influence on the ultimate state of the vessel and a coupling between the two conditions of heave and heave velocity can significantly alter the location of the vessel on such an integrity surface indicating a dramatic shift in vessel safety. Only generally can we say that maximum integrity values occur when heave velocity initial conditions are near zero for all heave.

3 Conclusions This paper presents the results of a thorough computational study into the interdependence of initial conditions in multiple degrees of freedom on predicting ultimate state of a simple box barge. While the hydrodynamic model is highly simplified it serves as a useful tool having demonstrated the influence coupled degrees of freedom can have on capsize prediction. The final two figures, 7 and 8, demonstrate the dependence of vessel stability on heave, heave velocity, sway and to a lesser extent sway velocity. While many models used in the present day literature focus upon one degree of freedom or a reduced order model, this work has shown that such models can be vulnerable to neglecting crucial dynamics. It is not sufficient to consider simply one state variable such as roll angle nor two state variables models coupling roll with roll velocity. This work demonstrates that elaborate couplings of multiple state variables in roll, sway and heave can significantly alter predictions of safe versus capsize and thus one must exercise care in the use of reduced order models.

Acknowledgements The authors would like to acknowledge the support of the National Defense Science and Engineering Graduate Fellowship program for sponsoring this research. Additionally the authors would like to thank Young-Woo Lee and Michael Obar whose work laid the basis for this paper.

Sway/Sway Velocity Initial Conditions on Capsize Modeling 191

Fig. 7 Sway/sway velocity integrity surface. Heave and heave velocity 0 at time of release, to. en≈3, h/=0.019.

Fig. 8 Heave/heave velocity integrity surface. Sway and sway velocity 0 at time of release, to. en≈3, h/=0.019.

192 L.S. McCue and A.W. Troesch

References Belenky VD, Liut KW, Shin YS. Nonlinear ship roll simulation with water-on-deck. Proc of the 2002 Stab Workshop. Webb Inst. 2002 Huang J, Cong L, Grochowalski S, Hsiung CC. Capsize analysis for ships with water shipping on and off the deck. Twenty-Second Symp on Naval Hydrodyn. Washington D.C. 1999 Lee YW. Nonlinear ship motion models to predict capsize in regular beam seas. Doctoral dissertation, Dep of Naval Archit and Mar Eng. University of Michigan Ann Arbor MI 2001 Lee YW, McCue LS, Obar MS, Troesch AW. Experimental and numerical investigation into the effects of initial conditions on a three degree of freedom capsize model. J of Ship Res 2006 50:1 63-84 Murashige S, Aihara K. Coexistence of periodic roll motion and chaotic one in a forced flooded ship. Int J of Bifurcation and Chaos 8:3 1998 619-626 Obar M, Lee YW, Troesch A. An experimental investigation into the effects initial conditions and water on deck have on a three degree of freedom capsize model. Proc of the Fifth Int Workshop on the Stab and Oper Saf of Ships. September 2001 Soliman, Thompson. Transient and steady state analysis of capsize phenomena. Appl Ocean Res 1991 Taylan M. Static and dynamic aspects of a capsize phenomenon. Ocean Eng. 30: 2003 331-350 Thompson J. Designing against capsize in beam seas: Recent advances and new insights. Appl Mech Rev. 50:5 May 1997 Thompson JMT, de Souza JR. Suppression of escape by resonant modal interactions: in shell vibration and heave-roll capsize. Proc of the Royal Soc of London. A452 1996 2527-2550 Vassalos D, K Spyrou. An investigation into the combined effects of transverse and directional stabilities on vessel safety. Fourth Int Symp on the Stab of Ships and Ocean Veh. Naples, Italy. 519-527 1990

The Use of Energy Build Up to Identify the Most Critical Heeling Axis Direction for Stability Calculations for Floating Offshore Structures Joost van Santen GustoMSC the Netherlands

Abstract For offshore structures like semi submersibles and jack-ups, hydrostatic stability is to be determined for what is called the weakest axis, which is not necessarily the same as the longitudinal axis of symmetry of the structure. When allowing trim to take place, the determination of the critical axis is complicated as free trimming leads to multiple solutions regarding the position for a given heel angle. It will be shown that for a freely floating structure, looking at the increase in potential energy can be used to identify those axis directions which are critical as well as realistic. The theoretical results will be illustrated with detailed data obtained for a two typical offshore structures using a standard stability program.

Nomenclature     Cxy CoB CoG E GML

= = = = = = = = =

displacement (t) heel angle axis direction (twist angle) trim angle cross moment of the waterline plane for rotation around its centre ≈∫x.y.dA Centre of Buoyancy Centre of gravity potential energy divided by displacement metacentric height for trim at a particular heel angle

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_10, © Springer Science+Business Media B.V. 2011

193

194 J. van Santen

GMT Mx My VCB x xcob, ycob y

= metacentric height for heel at a particular heel angle = moment around the longitudinal horizontal axis divided by displacement = moment around the transverse horizontal y axis divided by displacement = vertical centre of buoyancy above the centre of gravity = horizontal axis along the initial heeling axis, forward positive = location centre of buoyancy relative to CoG = horizontal axis perpendicular to the x axis (right handed system)

1 Introduction Historically, determination of the stability offshore rigs is seen as extension of stability for ships. For ships the longitudinal axis is taken as the heeling axis, where trim is allowed to remove the trimming moment. Early on, the offshore industry has recognized that the most critical axis is not necessarily the longitudinal axis. So, the wording critical axis was introduced, but as a kind of heritage one also had to consider free trim. This directly leads to a problem. This problem is that a given position can be defined by infinite combinations of axis directions, heel angle and trim. Two angles suffice to uniquely define the position. Using three angles means that one is superfluous. In the 80’s several papers appeared introducing the use of pressure integration to replace the conventional way of slicing up the structure and calculation of the contribution of each slice, (Witz and Patel, 1985, van Santen, 1986). In the publication of van Santen (1986), the problems mentioned above were raised and a way to avoid them by using free twist was introduced. In this paper, also the concept of energy build up was touched upon. More recently (Breuer and Sjölund, 2006), the problem was looked at again and using the build up of energy was proposed as a solution. In the underlying publication, the increase in potential energy of a structure due to forced heel with and without trim will be looked at and examples will be shown. This paper deals with freely floating structures. Effects due to mooring or dynamic positioning are specifically excluded.

Axis Direction for Stability Calculations for Floating Offshore Structures 195

2 Example with a Barge For a rectangular barge (Figure 1) with a raised forecastle, we can construct the righting arm curves for a range of axis directions (without trim) as shown in Figure 2. 25.0 m

Volume 25200 m3 LCG 70.0 m TCG 0.0 m VCG 17.0 m Intact draft 5.0 m Seawater density 1025 kg/m3

8.0 m

8.5 m

140.0 m

36.0 m

Fig. 1 Barge dimensions

40 90

35

80

Arm (m)

30 25 60

20 15

40

10 5

0

0 0

10

20

30

Heel angle (deg)

Fig. 2 Righting arm curves, no trim

Applying free trim has dramatic consequences as is shown in Figure 3. For some axis directions the righting arm curves stop prematurely. Why is this? In order to analyze this, the trimming moment as a function of trim angle is to be studied, see

196 J. van Santen

Figure 4. The trim angle for which this moment is zero is the free trim angle belonging to a particular heeled situation.

Fig. 3 Righting arm curves, free trim

Trimming arm for an axis direction of 80 deg zero value indicates a free trim position Trimming arm (m)

6

-50

Heel angle

7° No solution

6° 5° 4° 3°

7 3

6 5 4 3

0 -40

-30

Unstable solution

-20

-10

0

Stable solution -3

Trim angle (deg)

Fig. 4 Trimming arm curves

For an axis direction of 80 deg, when heeled beyond 5.6 deg heel, the zero crossings disappear, meaning that there are no solutions with zero trim moment. When passing a certain position, continuation of the righting arm curve can only

Axis Direction for Stability Calculations for Floating Offshore Structures 197

be achieved by reducing the heel whilst at the same time increasing the trim angle. This leads to the righting arm curve as shown in Figure 5 and the trim angle as shown in Figure 6. 15 Continuous path 10

in ble

trim

Righting arm

Sta

5 0 0

1

2

3

-5 n tr

i ble sta Un

-10

4

5

6

im

-15 Heel angle (deg)

Fig. 5 Continuous righting arm curves, free trim 0 0

1

2

3

4

5

6

-5

Trim angle [deg]

-10 -15 -20

Continuous path

-25 -30 -35 Heel angle (deg)

Fig. 6 Trim angle for continuous heel

For intact offshore rigs, some authorities require a range of positive stability of 30 (or 36) deg. For a longitudinal axis direction this criterion would be met, but for an axis direction of 80 deg, the structure would fail to meet this criterion. So, one could say that 80 deg axis direction is the most critical one and the rig would fail to comply with the range requirement.

198 J. van Santen

Looking at the structure (figure 1) it is obvious that one should select an almost longitudinal axis as the most critical axis direction and not 80 deg. But for a more generic structure it is not that easy to identify the critical axis direction. Why is it that from visual observations we reject 80 deg and accept 0 deg ? The key can be found in the rate at which the potential energy in the system increases with heel. For 0 deg axis direction this is far less than for 80 deg. The potential energy is given by the negative of the vertical distance (VCB) between the centre of gravity CoG and the centre of buoyancy CoB.

Energy  VCB  displacement,

(1)

where E will be used to denote energy divided by displacement, so E = -VCB. At the equilibrium position the energy is taken as the reference value, E0. For a change in heel, the increase in energy is given as

E  E     E 0

(2)

 VCB0  VCB   

The input of energy is the work done by the overturning heeling moment given by 

 M   d

(3)

0

where M() is the overturning moment divided by displacement. This results in: 

  VCB     VCB0    M   d 0



    righting arm  d 0

(4)

Note that for a free floating structure, a positive overturning moment results in a positive righting arm, but this means in fact that the counteracting restoring moment given by ycob is negative. The most critical axis can be viewed as the axis direction for which a given heel angle is reached with the least effort. In this way, the least energy is to be fed into the system in order to reach that particular heel angle. Figure 7 shows the amount of energy fed into the barge depending on axis direction and heel angle.

Axis Direction for Stability Calculations for Floating Offshore Structures 199

Fig. 7 Energy surface for the barge

The trim is fixed to zero. From this figure it is seen that for an axis direction of about 0 deg the lowest amount of energy is needed to reach a given heel angle. This agrees with the subjective feeling that the most critical axis is almost longitudinal. The problem now is how to determine the axis direction for which the energy increase is lowest. But before we do so, another example will be given, as found in the publication by Breuer and Sjöland (2006), see also Figure 8.

3 ABS Jack-Up Following discussions between MSC and ABS on the approval of a particular MSC jack-up, related to the Range of Stability (RoS) requirement (ABS rules 2008), Breuer and Sjölund (2006) published their paper in which they showed energy contour plots for a range of trim and heel angles. For the damage as indicated in Figure 8, the increase in energy depending on axis direction and heel angle is given in Figure 9. This figure shows that the lowest rate of increase is found for an axis direction of about 320 deg.

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Fig.8 Dimensions of the studied jack

Fig. 9 Energy surface of the studied jack up, damaged condition

For this direction, the righting arm is shown in Figure 10 (free trim). Selecting another axis direction (280 deg) results in a vanishing righting arm curve, as shown in Figure 11. Note the large trim angles related to the vanishing arm curve. Using an axis direction of 280 deg may lead to the false impression that the RoS criterion is not. Actually it is met, but it is for an axis direction of 320 deg.

Axis Direction for Stability Calculations for Floating Offshore Structures 201 2.5 Righting arm (m)

2

Trim angle (deg) 1.5 1 0.5 0 0

5

10

-0.5

15

Heel angle

-1 RoS=7+1.5*2.6 deg=10.9 deg -1.5

Fig. 10 Righting arm and trim angle, axis direction 320 deg

20 Righting arm (m) 15

Trim angle (deg)

10

5

0 0 -5

2

4

6

8

10

Heel angle

Fig. 11 Righting arm and trim angle, axis direction 280 deg

4 Axis Convention When using heel, trim and twist, it is important to have a clear understanding of their meaning. For the remainder of this publication, the following convention is used when positioning a structure in a heeled condition (see Figure 12):

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  

first the heeling axis direction is chosen, the structure is heeled around this axis the structure is trimmed around a transverse axis which was initially horizontal but changes direction due to heel.

5 Free Trim Versus Free Twist As is seen above, looking at energy increase during heel is a clear indicator for the determination of the critical axis direction. The next question is how to deal with trim. For ships, introducing free trim is clearly meant to obtain the lowest righting arm. This is easily seen when going from the free trim to the situation with nil trim. When going from the trimmed to the free trim situation, energy is dissipated, thus the righting arm is less than for the fixed axis direction and zero trim situation. The question is if this also works when varying the axis direction. For this purpose consider a heeled position with both a zero trimming moment and a zero moment around the twist axis which is the initially vertical axis. When applying a small change in the trim angle (d) and in the axis direction (d) reactive moments will result. For a constant displacement, using conventional hydrostatic considerations, these moments can be determined by looking at the horizontal and vertical components of the rotations. The horizontal component () causes a change in waterline shape and in CoB position. The combined effect translates into a reactive moment -GML . The rotation () around the vertical axis causes a transverse movement of the CoB which causes a reactive moment ycob . Thus, for a change of trim, d, the externally imposed moment needed for this change in trim becomes:

Axis Direction for Stability Calculations for Floating Offshore Structures 203 Select axis direction

Shift to CoG

z

axis system through CoG

y

x

Axis direction

Heel angle input axis system

Heel angle

Position after structure is heeled around a horizontal axis

z

y

Mx

x

My Heel angle

waterline

Trim angle

Position after structure is trimmed around an inclined transverse axis

z

x

y

Fig. 12 Application of twist, heel and trim

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My t   GM L cos   y cob sin   d

(5)

Note that the value of GM is the instantaneous metacentric height for the given heeled position. Similarly, due to a change of axis direction d the externally imposed moment becomes:

My a   GM L sin   y cob cos   d 

(6)

These moments are around a horizontal axis. When considering energy input, the moment is to be taken along the axis around which the rotation takes place. For instance, for trim, the moment around the trim axis is Myacos. Thus, for trim the energy input starting from the free twist, zero trim, position is:

1 dE t  cos   GM L cos   y cob sin   d2 2

(7)

Due to a change in axis direction the energy input starting from free twist is:

1 dE a  sin   GM L sin   y cob cos   d 2 2

(8)

The term between the brackets in Equation (7) is in general positive. In equation 8, for small heel angles the term between the brackets can be approximated by GML  - GMT  cos. So, for small heel angles, it is in general small but also positive. For larger angles it is in general positive. So, in general, when starting from the free twist position with zero trim and applying either a small change in twist or in trim an increase in energy in the system is found. Thus, the free twist position is the position with the lowest potential energy for a given heel angle. The question arises if there is a reduction in energy when going from a free twist position with zero trim to a new position (with a change in twist of d) and letting the rig free to trim with angle d. The energy input needed to twist is given by dEa. By letting the rig free to trim, the energy recovered is – dEt. So the total energy input is dEa - dEt. The twist angle is obtained by imposing a moment on the rig. This moment is nullified by letting it free to trim. So:

Axis Direction for Stability Calculations for Floating Offshore Structures 205

My a  My t

(9)

Using Equations (5) and (6) gives the relation between d and d:

d 

GM L cos   y cob sin   d GM L sin   y cob cos 

(10)

By substituting Equation (10) in Equation (8) the difference in energy input is found to be

dE  dE a  dE t GM L cos   y cob sin  1   d2 y cob 2 GM L sin   y cob cos 

(11)

Generally, the following applies:  GML is positive,  ycob is negative for a positive righting arm Thus, for the weakest axis, the numerator GML cos -ycob sin is in general positive. The denominator is also in general positive. So, when going from a free twist to a free trim situation the energy input is positive. This leads to the conclusion that the free twist situation contains less energy than fixing the axis direction combined with free trim. The free twist position equals the situation with a local minimum energy as both moments Myt and Mya are zero. For the jack-up shown in Figure 8, the numerical values for small variations in trim and twist around the free twist point are shown in Figure 13. These are based on the calculations with a stability program as well as on the approximations given above. Also, the calculated relation according to Equation (10) between axis and trim for which energy Ea is exchanged with Et is shown (“lowest envelope”). This figure shows the validity of the theoretical formulations. It is important to be aware that in the free twist method, trim is always zero. Hence, the heel angle is always equal to the steepest slope of an initial horizontal plane (like a deck).

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energy (m)

Energy depending on trim for varying axis directions damaged rig, 10 deg heel 0.005 0.004 0.003

325,3 324,3

0.002

323,3

0.001

322,3

SE-10 -1

-0.5

-0.001

Approximation axis 324,3 deg 326,3

321,3 0

0.5

1

trim angle (deg)

Calculated lowest envelope

Fig. 13 Energy depending on trim

6 Change in Axis Direction Due to a Change in Heel Angle In section 5 it is shown that the free twist method results in the lowest amount of potential energy build up for a given heel angle. For a free twist situation, the moment around the initial vertical axis is zero. When making a small step in the heel angle, the change in axis direction can be estimated using this condition. For a constant displacement a heel increase d results in moment around the twist axis of -Cxy · d, where Cxy = cross moment of inertia of the waterline for a rotation around its centre of floatation. Making this equal but opposite to the moment due to a small change in twist (Mya, Equation (6)) the change in axis direction results

d 

C xy / volume GM L sin   y cob cos 

d

(12)

It is seen that the axis direction is influenced by both the metacentric height (GML) and the righting arm (which equals -ycob).

Axis Direction for Stability Calculations for Floating Offshore Structures 207

7 Stability of the Solution The free twist position is characterized by the twist moment Mya being zero. It is not necessarily a position with the lowest energy, but it can also be a position with the highest energy. For a given heel angle, there are several axis directions for which the energy is either minimum or maximum. The maximum is by definition an unstable position, the minimum is a stable position. Stability is indicated by the energy input Equations (8) (for twist) and (7) (for trim). If the term GMLsin+ycobcos is negative, the position is unstable in twist. For an intact structure at a small heel angle, the following approximations apply: ycob ~ -GMT  sin() ~ cos() ~ 1.0

stability term   0 means stable 

 GM L  GM T  

(13)

Clearly, for small angles the position is stable when heeling is around the axis with the smallest GM value. For larger angles, the actual values of GML and ycob are to be considered. When the structure is in free to trim, also the stability for trim should be looked at. For trim to be unstable, the term GMLcos-ycobsin is to be negative. Keeping in mind that -ycob is the righting arm, it is seen that for a positive righting arm this is the case as long as GML, being the metacentric height for the given heeled position, is positive.

8 Case Studies For two structures, analyses have been done into the increase in potential energy during progressive heeling. These structures are:

 

ABS jack-up, a simplified semi submersible

8.1 ABS Jack-Up Intact. This structure, see Figure 8 has been analyzed for both the intact and damaged condition. For the intact condition, Figure 14 shows the energy surface for a range of heel angles (0-24 deg) and a range of axis directions (0-360 deg).

208 J. van Santen

Fig.14 Energy surface intact

It is seen that for larger heel angles 3 axis directions are present following the local minimum path. These are 900, 2100 and 3300. The 210 and 330 deg directions are in fact identical due to the symmetry of the structure. From the graph, it is also seen that the 330 (or 210) degree yields the lowest energy increase. For angles exceeding about 6 degrees heel there are 3 maxima and 3 minima in the plot. What is not clear is that for smaller heel angles there are only 2 maxima and 2 minima. The extremes at around 35 deg and 145 deg disappear for small heel angles. This is shown in Figure 15 which shows the axis directions which have a zero twist moment depending on the heel angle.

Fig. 15 Axis directions for minimum energy build up, intact

Axis Direction for Stability Calculations for Floating Offshore Structures 209

Damaged. The damage case has a damaged compartment as indicated in Figure 8. For a VCG of 23.45 m, the range of stability criterion of ABS is just met. This range should be 7 deg + 1.5 x steady heel (2.59 deg) = 10.89 deg. The plot of the energy (Figure 9) shows a minimum value for an axis direction of around 320 deg. Figure 16 shows the axis direction for which free twist is satisfied, i.e. where the moment around the y axis is indeed zero. The figure also shows the estimated axis direction based on Equation (12). In this estimate, the start value at zero heel is taken from the calculations with the stability program. The other values are based on the summed increments. A very good fit is seen between the estimated and the actual axis directions. For the damaged case, at larger heel angles, there are two paths following a local extreme in the energy surface. For small heel angles there is only a single path. 18 16

Heel angle (deg)

14 12

Unstable 10

Stable

8 6

Estimated path (eq13)

4

Minimum energy path

2 0 200

250

300

350

Axis direction (deg)

Fig. 16 Axis directions for minimum energy build up, damaged

8.2 Semi-Submersible Figure 17 shows a semi-submersible which at the given displacement has an intact draft of 9.106 m. Both the intact and damage condition pose a challenge when calculating the stability. Intact. Figure 18 shows the energy surface plot for the intact condition. Note that because of symmetry, the data for an axis direction of α is also found for axis direction α + 180. It is seen that for moderate heel angles, the minimum energy is

210 J. van Santen

found for an axis direction of 180 deg (and 0 deg). This is even better seen in the righting arm plot, Figure 19. When for 180 deg axis direction, heel is increased beyond 25.5 deg, the path follows a local unstable maximum instead of a local stable minimum. Instead, for larger heel angles, the minimum energy is found for axis directions of about 135 and 225 deg. A more detailed analysis can be done by looking at the twist moment depending on heel and axis direction.

Fig. 17 Shape of the semi submersible

Fig. 18 Energy surface, intact

Axis Direction for Stability Calculations for Floating Offshore Structures 211

Fig. 19 Surface plot of righting arm

Fig. 20 Surface plot of the trimming arm

Figure 20 shows this as a surface plot of xcob versus heel and axis direction. The free twist locations are those for which xcob equals 0. The combination of heel and axis directions for which the twist moment (or xcob) is zero are identified by the change in color in Figure 20. The stable and unstable positions can be identified by looking at the slope dxcob/ d for a given heel angle. Using this figure, the stable and unstable paths for which there is an extreme in the energy build up are as given in Figure 21 for axis directions between 120 and 240 degrees.

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Intact, VCG=40 m 50 Heel angle (deg)

Stable branches

Unstable branches

40 30 20 10 0 120

170 Axis direction (deg)

220

Fig. 21 Stable and unstable paths

Starting with zero heel, a gradual increase in the overturning moment will result in the rig to follow a seemingly erratic path: heel axis direction 0 – 1.6 180 deg 1.6 – 4.6 151-154 deg or 209 – 206 deg 4.7 – 25.4 180 deg 25.5 –50 133-138 deg or 227 – 222 deg At hindsight, the unstable area for very small heel angles is also seen in Figure 19 by a barely noticeable hill for axis 180 and heel 4 deg. Damaged. When damaging the rig by removing the pontoon corner part as indicated in Figure 17, the energy surface looks more simple than for the intact rig. The axis direction for minimum energy increase is around 140 deg, see Figure 22. Still, also in this case, the axis direction changes considerably for increasing heel as shown in Figures 22 and 23 When actually increasing the heel angle in a stepwise manner, the rig will suddenly change position when passing the 12 and 23 degrees heel, see Figures 23 and 24. At 12 degrees heel, it will change from axis direction about 140 deg to about 166 degree. At 23 degree heel it will change back to about 130 deg axis direction. Further study showed that this behavior hardly depends on the VCG.

Axis Direction for Stability Calculations for Floating Offshore Structures 213

Fig 22 Energy plot, damaged 40

Heel angle (deg)

35

Jump in axis direction for increasing heel

30 25 20 15 10

Jump in axis direction for decreasing heel

5 0 120

130

140

150

160

170

Axis direction (deg) Fig. 23 Relation between heel angle and axis direction, free twist

180

214 J. van Santen 5 4.5

Jump in righting arm

Righting arm (m)

4 3.5 3 2.5 2 1.5 1 0.5 0 0

10

20

30

40

Heel angle (deg)

Fig. 24 Righting arm following free twist

9 Use of Minimum Energy Path in Evaluating Stability Criteria When evaluating stability criteria, a distinction can be made between those without and with external influence. An example of the first group is ABS’s range of stability criterion for jack-ups. An example of the second group is the well known requirement on ratio between the area under the restoring moment and the wind overturning moment curves. For the first group, the minimum energy path can be followed as being the most critical, i.e. it requires the least effort to reach a particular heel angle. For the second group, both the magnitude of the restoring moment and of the overturning moment is to be considered. This highly complicates the task of selecting the most critical heeling axis direction. When following the minimum energy path, it is relatively easy to adapt the wind overturning moment to the instantaneous axis direction. But, this still assumes that the direction of the wind overturning moment is the same as that of the restoring moment. In general, it is well possible that this is not the case and that the wind overturning moment has a trimming component as well. But, this raises the question if one should absorb this by letting the unit trim or if the axis direction should be modified. It is also possible that the wind overturning moment for other directions is higher and thus more governing. Another way would be to look directly at the ratio between energy inputs versus energy build up for all axis directions without considering trim. When including possible downflooding, the complexity of the calculation increases further.

Axis Direction for Stability Calculations for Floating Offshore Structures 215

Apart from the calculation difficulties, the major issue is the nature of the calculation. Traditionally, the effect of wind for an otherwise calm sea is looked at. The effect of waves and resulting motions can be important, especially for a damaged structure (van Santen, 1999). Also, the calculation of the wind overturning effect is usually simplified in that the reactive force is assumed to work at the lateral center of resistance and that the forces due wind and the reactive forces do not introduce a yawing moment. Including the effect of mooring or DP makes the analysis even more complex. In view of this, there seems to be no justification to focus on one detail whilst ignoring other effects which may be much more important. In the end, we should not forget that the criteria are quite abstract. Being abstract, they lend themselves to consistency and reproducibility. The end results are not so much influenced by the details of the method used. As such, they fulfill the requirement of a criterion of which the evaluation is clearly defined and can be reproduced independently. By Couser (2003) the difficulty in programming the calculation of the AVCG curves as limited by the various requirements was mentioned. The examples shown here indicate that apart from the vagueness in the criteria, the structure itself can cause problems. For the semi-submersible case at transit draft it is almost impossible to do a proper AVCG calculation. A possible way out is to use a fixed axis direction with nil trim. For a righting arm curve constructed in this way to be acceptable, it should show only modest trimming moments as this indicates that the energy depletion by letting it free to twist or trim is small.

10 Conclusions 1. Extending the free trim approach as used for ships to offshore units, in combination with a varying heeling axis direction, may lead to severe interpretation problems. 2. A free trim or free twist approach is sensible as this generally leads to the slowest build up of potential energy for increasing heel angle. Thus the lowest righting arm curve is achieved. A pre-requisite is that the rig remains stable in trim and twist for the relevant range of heel angles. 3. The free twist approach with zero trim leads to the lowest gain in potential energy during heeling. This is based on both theory and data obtained from direct calculations. Thus when left free to trim and free to twist, the result is that the twist angle varies and that the trim angle remains zero. 4. Application of the lowest gain in energy approach is an unambiguous way to define the most critical or weakest axis. 5. For the determination of the critical axis direction, other effects are to be considered as well. When wind overturning is in the criterion, axes other than the weakest based on energy, may have to be considered. Also when looking at openings, other axis directions have to be looked at.

216 J. van Santen

For these complex cases, it is strongly suggested to perform the calculations for any axis direction and with the trim fixed at zero.

Postscript Since the original publication in 2009, some errors have been removed. Most important is the change in width of the compartment in figure 17 from 7 to 6 m. This has an important influence on the results. Also later at the City University conference 2009, an additional paper was published (Santen 2009) looking into more detail at the various methods including the Steepest Descent Method as proposed by Breuer and Sjölund, 2006. It explains and analyzes the SDM, gives details and shows results obtained with this method. The author performed some tests with scale models of the jack up and semisubmersible which confirmed the findings reported in this paper in a qualitative way. I owe many thanks to Andy Breuer for having many open minded and fruitfull discussions about the problems met when doing stability calculations for offshore structures. His encouragement to publish my views resulted in this and in the City University paper.

References ABS rules for building and classing Mobile Offshore Drilling Units 2008. Breuer JA and Sjölund KG (2006) “Orthogonal Tipping in Conventional Offshore Stability Evaluations”, Proc of the 9 th Int conf on Stab of Ships and Ocean Veh (STAB 2006), Rio de Janeiro. Santen JA (1986) “Stability calculations for jack-ups and semi-submersibles”, Conf on computer Aided. Design, Manuf and Oper in the Mar and Offshore Ind CADMO 1986, Washington. Santen JA (1999) “Jack-up model tests for dynamic effects on intact and damaged stability”, Seventh Int Conf The Jack-up Platform: Design, Constr and Oper, City Univ, London, Sept. Santen JA (2009) The use of energy build up to identify the most critical heeling axis direction for stability calculations for floating offshore structures, review of various methods. 12th Jack-Up Conf 2009 City Univ, London. Witz JA, Patel MH (July, 1985) “A pressure integration technique for hydrostatic analysis”, RINA suppl pap, Vol. 127. Couser P (2003) “A Software Developer’s Perspective of Stability Criteria”, 8th Int Conf on the Stab of Ships and Ocean Veh, STAB 2003, Madrid.

Some Remarks on Theoretical Modelling of Intact Stability N. Umeda*, Y. Ohkura*, S. Urano*, M. Hori* and H. Hashimoto* *Dept. of Naval Architecture and Ocean Engineering, Osaka University, 2-1, Yamadaoka, Suita, Osaka, 565-0871, JAPAN

Abstract It is essential for realising risk-based design or performance-based criteria to establish theoretical modelling of extreme behaviours of an intact ship in waves up to capsizing. For this purpose, this paper summarises latest progresses in the researches conducted by the authors at Osaka University for improving theoretical modelling of intact stability in waves. Firstly, the methodology for calculating capsizing probability in beam wind and waves with piece-wise linear assumption is examined. Here a new formula covering both capsizing towards windward and leeward directions is provided. Secondly, for calculating restoring moment in longitudinal waves with sufficient accuracy, a hydrodynamic prediction method with wave-making and lifting components taken into account is proposed, and is reasonably well compared with captive model tests of a post Panamax container ship in head and following waves. This can be used in place of the Froude-Krylov prediction or captive tests for assessing the danger of parametric rolling for a practical purpose.Thirdly, for more efficiently identifying dangerous conditions in following and quartering waves, a numerical method for determining a heteroclinic bifurcation is proposed with its successful example to applied to surfriding thresholds with nonlinear factors taken into account.

1 Introduction At the International Maritime Organisation (IMO), the review of Intact Stability Code started in 2002 and could open a door to use direct stability assessment with physical and theoretical modelling as an alternatives to prescriptive rules in the near future. Here quantitative accuracy of these modelling is essential. The benchmark testing of the 23rd International Towing Tank Conference (ITTC), however, revealed that existing theoretical models for predicting capsizing do have only qualitative accuracy. This means that the existing best models can predict whether a ship capsizes or not and how she does but they can not predict M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_11, © Springer Science+Business Media B.V. 2011

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218 N. Umeda et al.

exactly when she does after the specified initial conditions. (Vassalos et al., 2002) Therefore, improvement of theoretical modelling is indispensable for its practical use. After the 23rd ITTC, the research team of Osaka University continues to improve its own theoretical models from various aspects. (Umeda, Hashimoto et al., 2003, Umeda et al., 2004, Hashimoto et al., 2004A, Hashimoto et al., 2004B) In this paper, some of their latest progress are presented to initiate the discussion at the stability workshop. These theoretical modelling to be investigated at Osaka University covers the dead ship condition, parametric rolling and broaching. The dead ship condition is related to the revision of the weather criterion and the others were the capsizing scenarios that the 23rd ITTC benchmark testing specified.

2 Capsizing Probability in Beam Wind and Waves Although a ship could capsize in following and quartering seas with much smaller wave steepness than in beam seas, it is still important to assess stability in beam seas. This is because dangerous situations in following and quartering seas can be avoided by ship operation but those in beam seas cannot be. For example, if all operational means such as propeller thrust and rudder control are lost, a ship that is almost longitudinally symmetric suffers bean wind and waves. Here, since a ship cannot escape from a severe sea state by herself, she should survive for sufficiently long duration. For guaranteeing survivability of such dead ship condition (IMO, 2001), the weather criterion was adopted by several administrations (Yamagata, 1959) and then by IMO (IMO, 2002). Since this criterion utilises some empirical methods to predict aero- and hydrodynamic coefficients, applicability of this criterion to new ship types such as a large passenger ship is rather questionable. However, because the probabilistic safety level of the weather criterion was adjusted by selecting the average wind velocity with casualty statistics, (Yamagata 1959) it is necessary to evaluate the safety level that the current weather criterion implicitly guarantees for conventional ships. (Umeda et al. 1992, Umeda & Yoshinari 2003) For this purpose, analytical methods to calculate capsizing probability in beam wind and waves are indispensable. This is because numerical or experimental methods require prohibitively many realisations to obtain the reliable value of capsizing probability for a practical ship, which should have very small capsizing probability. Among some existing analytical methods, a piece-wise linear approach proposed by Belenky (1993) seems to be the most promising because it utilises analytically-obtained exact but simple solutions of linear equations only. The authors investigate Belenky’s method by executing numerical calculation with his exact and simplified methods and extend his methods to the case in beam wind and waves with both capsizing in windward and leeward directions taken into account. As a result, some problems and their measures are found. Among

Some Remarks on Theoretical Modelling of Intact Stability 219

them this paper shows one problem that capsizing probability could exceed 1 when a sea state is extremely severe. According to Belenky (1993), the formula of capsizing probability for the duration T without beam wind is as follows:

PT   2  [1  exp(u l T )]  PA  A  0;    mo 

(1)

where

A







1 1  p1  2  p1  1  1  2

(2)

: apparent roll angle, m0: border of two roll angle regions, 1, 2: eigenvalues of the roll system in the second range, 1. 1 : initial values of roll angle and roll angular velocity in the second range, p1, p1 : initial values of forced roll angle and roll angular velocity in the second range, ul: expected number of up-crossing at mo for a unit time and PA: probability of positive value of the coefficient A when the up-crossing of  occurs. On the other hand, the authors propose the following formula for the case of wind and waves.

PT   1  exp[{ul PA  A  0;   mo   u w PA  A  0;   mo }  T ]

(3)

where uw is expected number of down-crossing at -mo for a unit time. If we ignore wind here, the following formula can be obtained:

PT   1  exp[2u l PA  A  0;   mo   T ]

(4)

Therefore, P(T) defined here cannot be greater than 1. The authors applied the above new formula to a car carrier, whose principal particulars are shown in Table 1. Numerical results are presented in Fig. 1. Here the righting arm is approximated with two lines by keeping the metacentric height, the angle of vanishing stability and the dynamic stability from upright to the angle of vanishing stability because a time-varying external force induces capsizing in this scenario. The wind velocity, UT, is assumed to be constant and waves are done to be fully developed with this wind velocity and to be modelled with ITTC spectrum. PA are calculated with both exact and simplified methods. The former is derived from 3-dimensional Gaussian probabilistic density function; the latter is based on the assumption that no resonance occurs in the second region. The results demonstrate that capsizing probability tends to 1 when the sea state becomes

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Capsizing Probability

extremely severe and Belenky’s simplified method slightly overestimates capsizing probability obtained by the exact method. Further discussion is available in the work of Kubo et al. (2010). Moreover, the effect of fluctuating wind has been incorporated for the capsizing probability calculation by the authors. (Francescutto, Umeda et al. 2004)

15.00 1.00E+00 1.00E-04 1.00E-08 1.00E-12 1.00E-16 1.00E-20 1.00E-24 1.00E-28 1.00E-32 1.00E-36

25.00

35.00

45.00

U (m/s) 55.00

Simplified Method Exact Method

Fig. 1 Capsizing probability of the car carrier in beam wind and waves by using the simplified and exact methods of the piece-wise linear approach. (Paroka Ohkura & Umeda 2006)

Table 1. Principal dimensions of the car carrier

Items

Car carrier

Length overall: Loa

190 m

Length between perpendiculars: Lpp

180 m

Breadth: B

32.20 m

Draft: T

8.925 m

Vertical centre of gravity: KG Metacentric height: GM Lateral projected area: AL Height to centre of lateral projected area: HC

14.105 m 1.300 m 4327.860 m2 11.827 m

Some Remarks on Theoretical Modelling of Intact Stability 221

3 Hydrodynamic Effect on the Change in GM Due to Waves At the 23rd ITTC benchmark testing even the most reliable mathematical model can provide only qualitative agreements with free-running model experiments for capsizing due to parametric rolling. As well known, parametric rolling is induced by the change in GM due to waves. Thus the accurate prediction of the change in GM due to waves is essential but all mathematical model in the benchmark testing predict the change in GM due to waves with the Froude-Krylov assumption. The captive model experiment for the ITTC A-1 Ship, as a follow-up of the ITTC benchmark testing, revealed that the Froude-Krylov calculation significantly overestimates the change in GM due to waves. (Umeda et al., 2004) Possible reason of the difference between the experiment and the Froude-Krylov calculation are hydrodynamic forces such as a wave-making effect and a hydrodynamic lift due to a heeled hull. For modelling such hydrodynamic effects on the change in GM due to waves, Boroday (1990) attempted to use a strip theory with the heel angle taken into account. He reported that good agreement with model experiment is obtained by ignoring wave-making damping components but the wave-making damping deteriorates the agreement. For finding a final conclusion on this issue, the authors reformulate a strip theory to calculate the change in GM due to waves for a heeled ship in longitudinal waves. Here all radiation components in the roll moment due to heave and pitch motions and the heel effect of the roll diffraction moment are consistently taken into account; the end terms are also included to explain hydrodynamic lift components. Two-dimensional hydrodynamic forces are estimated with an integral equation method with the Green function. The subject ship used here is a 6600TEU postPanamax container ship, whose principal particulars are shown in Table 2. Its numerical results are shown together with the captive model test results in Fig. 2. While the Froude-Krylov calculation does not depend on the Froude number, the present calculation depends on the Froude number because of hydrodynamic effects. In following seas the Froude-Krylov calculation significantly overestimates the amplitude of GM variation but the present calculation well explains the reduction of the amplitude. In head seas the Froude-Krylov calculation overestimates the amplitude of the model runs with the Froude number of 0.1 but underestimates those with the Froude numbers of 0.2 and 0.3. The present calculation predicts the amplitude in the order obtained from the experiment. This improvement indicates that the hydrodynamic effect cannot be ignored when we accurately estimate the change in GM due to waves.

222 N. Umeda et al.

Fig. 2 Amplitude of GM variation in following and head seas for the post Panamax container ship with the wave length to ship length ratio of 1.0. Here h/ and Fn indicates the wave steepness and the Froude number, respectively. And “cal. (FK)” means the Froude-Krylov calculation.

Some Remarks on Theoretical Modelling of Intact Stability 223 Table 2. Principal dimensions of the post Panamax container ship

Items length : Lpp breadth : B depth : D draught at FP : Tf mean draught : T draught at AP : Ta block coefficient : Cb pitch radius of gyration : yy/Lpp longitudinal position of centre of gravity from the midship : xCG metacentric height : GM natural roll period : T 

Containert Ship 283.8m 42.8m 24.0m 14.0m 14.0m 14.0m 0.630 0.239 5.74m aft 1.08m 30.3 s.

4 Surf-Riding Threshold Broaching associated with surf-riding was also the capsizing scenario used in the 23rd ITTC benchmark testing for intact stability and quantitative disagreement with free-running model experiments was pointed out. In particular, the surf-riding threshold estimated with the existing model usually underestimates that from the experiment. Then some of the authors continue their effort to improve theoretical modelling by examining various nonlinear factors that had been ignored. As a result, they found that nonlinearity in the wave-induced surge force acting on a hull and the wave effect on propeller thrust are essential to improve the prediction accuracy of surf-riding threshold. (Hashimoto et al, 2004A) It is well established that the prediction of surf-riding threshold could depend on initial conditions if we simply apply numerical simulation in time domain but the surf-riding threshold for a self-propelled ship starting from sufficiently low propeller revolution does not depend on initial conditions. This is because onset of surf-riding or broaching can be regarded as a heteroclinic bifurcation. (Makov, 1969, Umeda, 1990, Spyrou, 1996, Umeda, 1999) That is, surf-riding or broaching occurs when an unstable invariant manifold of a saddle-type equilibrium in a phase space coincides with a stable invariant manifold of neighbouring saddletype equilibrium. Although a perturbation method (Ananiev, 1966 ) and an exact method (Spyrou, 2001) are available for a simplest mathematical model, the introduction of nonlinear factors and coupling could prevent their application. A numerical technique to identify surf-riding threshold as a heteroclinic bifurcation can be applied to more complicated theoretical model. The technique reported so far, however, is not suitable for a computer algorithm. (Umeda, 1990, Umeda, 1999) Here it is necessary to calculate time series starting from a saddle-type equilibrium

224 N. Umeda et al.

to other saddle-type one with the information from locally linearised systems. Thus, in this paper, a mathematical procedure suitable for a computer program is proposed with a successful example for following sea case and perspectives towards quartering sea case is added. For a simplicity sake, the uncoupled surge motion in following waves is investigated with the following mathematical model.

(m  mx )G  T (u ,  G ;  a , n)  R (u )  X W ( G ;  a )

(5)

where G: horizontal displacement of the ship centre to a wave trough, m: ship mass, mx: ship added mass, u: ship velocity ( G  u  c ), n: propeller revolution number, a: wave amplitude, T: propeller thrust, R: ship resistance and Xw: waveinduced surge force acting on a hull. The proposed procedure can be summarised as follows. First, by assuming a certain value of n, the equilibria from Equation (5) are identified and their eigenvalues and eigenvectors of the locally linearised systems at the equilibria are calculated. Here the equilibria correspond to surf-riding; one is stable and the other is unstable. Then we integrate Equation (5) with time from the point that is slightly shifted from an unstable equilibrium in the unstable direction of its eigenvector. And we also integrate Equation (5) with time backwards from the point slightly shifted from the neighbouring unstable equilibria in the stable direction of its eigenvector. Then we quantify the difference between two obtained values of u at a specified value of G as (n). Next, we apply the Newton method to find the parameter n* that  (n*)  0 . The obtained n* can be regarded as a heteroclinic bifurcation point. By using the relationship between the propeller revolution and ship speed in calm water, the nominal Froude number as the bifurcation point is determined. The above numerical method is used to determine surf-riding threshold for the ITTC A-2 ship with and without the nonlinear surge force and the wave effect on propeller thrust taken into account. Figs. 3-4 illustrate rapid convergence of the Newton method applied to their problem. The results shown in Fig. 5 show that the nonlinear surge force and the wave effect on propeller thrust significantly increase critical velocity for surf-riding. This means that these two nonlinear factors are essential to improve prediction accuracy for capsizing associated with surf-riding.

Some Remarks on Theoretical Modelling of Intact Stability 225

Fig. 3 Time domain simulation from saddle-type equilibria with time forwards and backwards with the wave steepness of 1/15 and the wave length to ship length ratio of 1.5. Here c and  are wave celerity and wave length, respectively: Fn indicates the nominal Froude number.

Fig. 4 Convergence of the procedure for determining the surf-riding threshold. Here the wave steepness is 1/15 and the wave length to ship length ratio is 1.5.

For predicting the broaching threshold, it is necessary to use an coupled surgesway-yaw-roll-rudder motion. As a result, the stable invariant manifold becomes 7-dimensional while the unstable invarioant manifold is 1-dimensional. (Umeda, 1999) Thus the additional condition to be satisfied is that the vector normal to the stable invariant manifold should be orthogonal to the unstable invariant manifold. Numerical works based on this procedure were published later (Umeda et al., 2007)

226 N. Umeda et al. 0.45 0.40

present method conventional method

Fn 0.35 0.30 0.25 0.8

1.0

1.2

1.4 l/L

1.6

1.8

2.0

Fig. 5 Surf-riding threshold for the ITTC A-2 ship in following seas predicted by numerical procedure to identify a heteroclinic bifurcation with the wave steepness of 1/10. Here Fn and /L mean the nominal Froude number and the wave length to ship length ratio, respectively. The present method is one that takes nonlinearity in wave-induced surge force and the wave effect of propeller thrust into account and the conventional method ignores them.

5 Conclusions As follow-ups of the 23rd ITTC benchmark testing, the research team of Osaka University provides some improvements in theoretical modelling of intact stability from the viewpoints of probabilistic theory, hydrodynamics and nonlinear dynamics. The major outcomes are as follows: 1. The formula to calculate capsizing probability in a dead ship condition is presented as an extension of Belenky’s piece-wise linear approach. 2. Hydrodynamic modelling of the change in GM due to waves could improve agreements with captive model experiments. 3. A numerical algorithm to determine the surf-riding threshold is presented with some important nonlinear elements taken into account. The newly-obtained results provide much larger critical velocity for surf-riding than the conventional results.

Acknowledgements This work was carried out as a research activity of the RR-S202 and RR-SP4 research panel of the Shipbuilding Research Association of Japan in the fiscal years of 2003 and 2004, funded by the Nippon Foundation. The work is partly supported by a Grant-in-Aid for Scientific Research of the Ministry of Education, Culture, Sports,

Some Remarks on Theoretical Modelling of Intact Stability 227

Science and Technology of Japan (No. 15360465). The authors express their sincere gratitude to the above organizations. The authors also thank Dr. Gabriele Bulian for his effective advice on the formula of capsizing probability, which enabled us to correct Equations (3) and (4) in this version.

References Ananiev DM (1966) “On Surf-Riding in Following Seas”, (in Russian) Transactions of Kryloc Soc, 13:169-176. Belenky VL (1993) “A Capsizing Probability Computation Method” J of Ship Res, 37:3: 200-207. Boroday IK (1990) “Ship Stability in Waves: On the Problem of Righting Moment Estimations for Ships in Oblique Waves”, Proc of the 4th Int Conf Stab of Ships and Ocean Veh, Naples, II: 441-451. Francescutto A, N Umeda et al (2004) “Experiment-Supported Weather Criterion and Its Design Impact on Large Passenger Ships”, Proc of the 2nd Int Maritime Conf on Design for Safety, Osaka Pref Univ, pp. 102-113. Hashimoto H, N Umeda and A Matsuda (2004A) “Importance of several nonlinear factors on broaching prediction”, J of Marine Sci and Technol, 9: 2: 80-93. Hashimoto H, N Umeda and A Matsuda (2004B) “Model Experiment on Heel-Induced Hydrodynamic Forces in Waves for Broaching Prediction”, Proc of the 7th Int Workshop on Stab and Oper Safety of Ships, Shanghai Jiao Tong Univ, pp. 144-155. IMO (2001) “SOLAS : consolidated edition, 2001”, 44. IMO (2002) “Code on Intact Stability for All Types of Ships Covered by IMO Instruments”. Kubo T, E Maeda and N Umeda. (2010). “Theoretical Methodology for Quantifying Probability of Stability Failure for a Ship in Beam Wind and Waves and its Numerical Validation”, Proceedings of 4th International Maritime Conference on Design for Safety, Fincantieri, pp. 1-8. Makov Y (1969) “Some Results of Theoretical Analysis of Surf-Riding in Following Seas” (in Russian) Transactions of Krylov Soc, 126: 124-128. Paroka D, Y Ohkura and N Umeda (2006) “Analytical Prediction of Capsizing Probability of a Ship in Beam Wind and Waves”, J of Ship Res, 50:187-195. Spyrou KJ (1996) “Dynamic Instability in Quartering Seas: the Behaviour of a Ship During Broaching” J of Ship Res, 40: 46-59. Spyrou KJ (2001) “Exact Analytical Solutions for Asymmetric Surging and Surf-Riding”, Proceedings of the 5th Int Workshop on Stab and Oper Safety of Ships, Univ of Trieste, 4:4:1-3. Umeda N (1990) “Probabilistic Study on Surf-riding of a Ship in Irregular Following Seas”, Proc of the 4th Int Conf on Stab of Ships and Ocean Veh, Naples, I: 336-343. Umeda N (1999) “Nonlinear Dynamics on Ship Capsizing due to Broaching in Following and Quartering Seas”, J of Marine Sci of Technol, 4: 16-26. 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”, J of Marine Sci and Technol, 7:145-155. Umeda N, H Hashimoto, D Vassalos, S Urano and K Okou (2004) “Nonlinear Dynamics on Parametric Roll Resonance with Realistic Numerical Modelling”, Int Shipbuilding Prog, 51: 2/3: 205-220. Umeda N, Y Ikeda and S Suzuki (1992) “Risk Analysis Applied to the Capsizing of High-Speed Craft in Beam Seas”, Proc of the 5th Int Symposium on the Practical Design of Ships and Mobile Units, Elsevier, 2:1131-1145.

228 N. Umeda et al. Umeda N and K Yoshinari (2003) “Examination of IMO Weather Criterion in the Light of Capsizing Probability Calculation, Conf Proc of the Soc of Naval Archit of Japan, 1: 65-66. Umeda N, M Hori and H Hashimoto (2007), “ Theoretical Prediction of Broaching in the Light of Local and Global Bifurcation Analysis”, Int Shipbuilding Prog, 54:4, 269-281. Vassalos D et al (2002) “The Specialist Committee on Prediction of Extreme Ship Motions and Capsizing –Final Report and Recommendations to the 23rd ITTC”, Proc of the 23rd ITTC, INSEAN (Rome) II: 611-649. Yamagata M (1959) “Standard of Stability Adopted in Japan”, Transactions of the Inst of Naval Archit, 101: 417-443.

3 Parametric Rolling

An Investigation on Head-Sea Parametric Rolling for Two Fishing Vessels Marcelo A.S. Neves*, Nelson A. Pérez**, Osvaldo M. Lorca***, Claudio A. Rodríguez* *LabOceano, COPPE/UFRJ, Brazil **Universidad Austral de Chile ***InterMoor Brazil

Abstract

The paper describes an experimental and numerical investigation on the relevance of parametric resonance for two typical fishing vessels in head seas. The investigation is aimed at assessing the influence of the incorporation of a transom stern to the design. Results for different Froude numbers are discussed. The first region of resonance is investigated. Distinct metacentric heights and wave amplitudes are considered. Quite intense resonances are found to occur. These are associated with specific values of metacentric height, ship speed and stern shape. In order to analyse the experimental/numerical results, analytic consideration is given to distinct parameters affecting the dynamic process of roll amplification. The influences of heave, pitch, wave passage effect, speed and roll restoring characteristics are discussed.

1 Introduction Parametric resonance of ships in longitudinal regular waves is discussed in the paper, as this is an important particular situation. Results for two typical fishing vessels are compared. Head seas conditions are contemplated in the discussions. Recently, some papers have called attention to the problem of parametric excitation in head seas. (France et al. 2003) reported on strong roll amplification in the case of a container ship. (Dallinga et al. 1998) and (Luth and Dallinga 1999) reported on the development of head seas parametric resonance in cruise vessels. Previous investigations conducted by the authors, Pérez and Sanguinetti (1995), Neves, Pérez and Valerio (1999), indicated that in longitudinal regular waves, at zero speed of advance, resonant parametric amplification may result in large and dangerous rolling motions, particularly for fishing vessels with a transom stern in low metacentric height loading conditions. In the present paper, experimental results for different Froude numbers in head seas are discussed. This is a condition often encountered by fishing vessels, as while trying to maintain position in rough weather. M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_12, © Springer Science+Business Media B.V. 2011

231

232 M.A.S. Neves et al.

For the present study, comprehensive testing of two typical fishing vessels has been carried out, aiming at investigating the relevance of parametric resonance of the roll motion for the two ships in head seas under distinct speeds. In the experiments, the first region of resonance was investigated. This is defined as the condition corresponding to the encounter frequency coinciding with twice the roll natural frequency. The two vessels have very similar characteristics but their sterns are different. One is a round stern vessel; the other one has a transom stern. For each hull, two metacentric heights were tested. Additionally, different speeds and wave amplitudes were considered. The transom stern configuration had already shown large roll amplifications in the case of zero speed of advance (Neves et al. 1999). One relevant question that the present paper intends to answer is whether a transom stern arrangement defines strong parametric resonance when different speeds in head seas are considered. Some conditions show intense amplification. The influence of hull stern shape is discussed. Its influence on the dynamics of roll parametric resonance is shown to be relevant when low metacentric height conditions are considered. Influence of damping level is also discussed. For some tuning conditions the transom stern hull may display at high speeds stronger roll amplifications than those registered in the low speed range. Results are interpreted having into account the main terms affecting the energy balance of coupled modes. For this purpose, the distinct influence of the heave and pitch motions are discussed. The paper shows that similar hulls tested in similar conditions may display very distinct responses at specific testing conditions. The experiments clearly demonstrate that in some cases strong parametric resonance in head seas can take place in quite few cycles. It is demonstrated that the effect of speed on parametric resonance is strongly dependent on stern shape. A transom stern, incorporating longitudinal asymmetry in flare, may exert a significant influence in establishing the tendency of a fishing vessel hull to display strong parametric amplification in head seas, particularly in a condition of low metacentric height. These conclusions are relevant in practice for hull design and operational considerations in rough seas. A coupled mathematical model capable of describing roll parametric amplification in waves is introduced. The connection between the coupling terms and parametric excitation is discussed. An assessment is made of the influence of some relevant parameters directly related to practical aspects of ship design for improved stability. Finally, the dynamics of motion around one particular tuning frequency is analysed. Considering steady roll responses around the ωe  2ωn tuning condition for a given reference speed, it is observed that for speeds slightly lower than this reference speed, due to reduced damping and non-linear restoring characteristics of the hull, roll responses are larger than at the exact tuning. It is shown that the roll response curve has a marked asymmetry with respect to the reference speed. Jump effect is identified as a potentially relevant dynamic characteristic to be taken into account in the design of ships subjected to roll parametric amplification.

An Investigation on Head-Sea Parametric Rolling for two Fishing Vessels 233

2 Tested Conditions The tests were conducted at the Ship Model Basin of the Austral University, Valdivia, Chile. The tank main dimensions are: 45.0m of total length, 3.0m of breadth and 1.8m of depth. A flap-type wave generator positioned at one extreme of the tank generates regular waves. More details about the towing tank and experimental arrangements may be encountered in (Neves et al. 1999). Whenever a ship model is to be towed in waves, a major difficulty has to be overcome: to perform the towing of the model without undesirable interference of the towing arrangement with the wave excited bodily motions of the model. This is particularly complicated when large perturbations are associated with the translational average motion of the hull. The satisfactory solution to the problem that was implemented in the present series of tests is illustrated in Fig. 1. Two auxiliary lines were respectively fixed to the bow and stern of the model at calm water level, with these two lines tied to the towing wire. The resulting elasticity of the set was found in all cases to be appropriate in order to secure free evolution of the different (symmetric and anti-symmetric) modes of motion at a controlled speed. In particular, large yaw motions could develop without noticeable interference in practically all runnings.

Fig. 1 Towing arrangement adopted in the experiments

The main characteristics of the ships used in this paper are listed in Table I. Figure 2 shows their lines plans. From Table I and Figure 2 it may be observed that the two hulls have very similar dimensions and characteristics but different stern arrangements. The first one will be called RS. It is a quite typical fishing vessel hull form with conventional round stern. The second hull, which will be called TS, corresponds to a typical transom stern fishing vessel. The tested models were built to a scale 1:30. As mentioned previously, the two hulls had already been tested under parametric resonance in the case of zero speed of advance, (Pérez and Sanguinetti 1995, Neves et al. 1999).

234 M.A.S. Neves et al. Table I Principal Particulars of Ships

Denomination Length (m) Length between perp. (m) Beam (m) Depth (m) Draught (m) Displacement (tons) Waterplane area (m2) Radius of gyration ryy (m)

RS 24.36 21.44 6.71 3.35 2.49 162.6 102.5 5.35

TS 25.91 22.09 6.86 3.35 2.48 170.3 121.0 5.52

Fig. 2 Body plans of tested vessels: RS (left); TS (right)

In the experiments, the first region of parametric resonance was investigated. This is defined as the condition corresponding to the encounter frequency coinciding with twice the roll natural frequency (Kerwin 1955, Paulling and Rosenberg 1959). In other words, the conditions were defined associated with one single encounter

An Investigation on Head-Sea Parametric Rolling for two Fishing Vessels 235

frequency e    kU cos  for each loading condition. For each hull two loading conditions were defined, one with low GM and the other one with high GM. Table II Tested Conditions for RS

GM

Fn

(m)

 (m)

0.10

39.0

0.14

43.2

0.20

49.2

0.34

62.4

0.10

28.7

0.14

32.1

0.20

37.0

0.34

47.8

0.34

0.48

a (m) 0.59 0.69 0.84 0.51 0.80 0.84 0.75 0.87 0.96 0.90 1.02 0.48 0.66 0.90 0.54 0.66 0.90 0.48 0.66 0.90 1.02 1.02

Wave steepness



1/33 1/28 1/23 1/42 1/27 1/26 1/33 1/28 1/26 1/35 1/31 1/30 1/22 1/16 1/30 1/25 1/18 1/39 1/28 1/21 1/18 1/23

(deg.) 7 14 30 3 20 24 5 7 17 0 5 10 20 32 9 18 30 5 16 20 21 0

hw / 

In the above definition,  is the wave frequency, U is the ship speed, k is the wave number and  is the angle of wave incidence with respect to the ship. For head seas,   180 0 . The tested conditions were those defined in Tables II and III. In these tables, wavelength.

a is wave amplitude, hw is the wave height and  is

236 M.A.S. Neves et al. Table III Tested Conditions for TS

GM (m)

Fn



(m) 0.11

31.6

0.15

35.1

0.20

39.4

0.30

47.6

0.11

24.5

0.15

27.5

0.20 0.30

31.2 38.1

0.37

0.50

a (m) 0.30 0.66 0.45 1.02 0.45 0.60 0.60 0.78 0.39 0.63 1.02 0.39 0.60 1.08 1.02 1.02

Wave steepness



1/53 1/24 1/39 1/17 1/44 1/33 1/40 1/31 1/31 1/19 1/12 1/35 1/23 1/13 1/15 1/19

(deg.) 15 27 18 28 4 19 5 38 19 22 27 2 13 16.5 0 0

hw / 

The last columns of Tables II and III present, for each speed and wave amplitude, for the tested conditions corresponding to e  2n (where  n is the natural frequency in the roll mode) the registered values of  (in degrees), the final (steady) roll amplitude observed in each test after all transients had died out. Clearly, for constant encounter frequency, this final roll amplitude is a measure of the level of parametric amplification in each tested condition. In summary, the experimental test programme involved two hulls with distinct stern arrangements to be compared, each vessel with two metacentric heights, varying speeds and wave amplitudes.

3 Equations of Motion A derivative model is adopted. Following the nomenclature previously used in (Neves and Rodríguez 2006) equations of motion in heave, roll and pitch, with hydrostatic and wave passage nonlinearities defined up to third order may be written as:

An Investigation on Head-Sea Parametric Rolling for two Fishing Vessels 237

m  Z z z  Z z z  Z   Z    Z z z  Z    1 Z zz z 2  1 Z   2  1 Z   2  Z z z 

2 2 2 1 1 1 1 1 1 Z zzz z 3  Z zz z 2   Z z  2 z  Z  2   Z z  2 z  Z   3  Z z t z  6 2 2 2 2 6 Z  t   Z z t z  Z zz t z 2  Z  t   Z z t z  Z  t  2  Z  (t ) 2  Z w t 

(1)

J



1  K    K    K      K     3  K    K z z  K    K zz z 2   2 1 1 3 2 K    K     K z z  K  t   K  t   K z t z  K  t   K w (t ) 6 2

xx

(2)

J

1 1 1  M    M    M z z  M z z  M z z  M    M zz z 2  M   2  M   2  2 2 2 1 1 1 1 1 1 3 2 2 2 2 M z z  M zzz z  M zz z   M z  z  M     M z  z  M   3  6 2 2 2 2 6 2 M z t z  M  t   M z t z  M zz t z  M  t   M z t z  M  t  2 

yy

M  (t ) 2  M w t 

(3) On the right hand side of these equations, Z w (t ) , K w (t ) and M w (t ) describe the wave external excitation in the heave, roll and pitch modes, respectively. These wave actions are assumed to linearly related to the wave amplitude, a. In the left hand side of the equations, nonlinear restoring terms include dependence on all body modes ( z ,  ,  ) and wave passage ( ) . Dots refer to velocities; double dots to accelerations. In all modes, coefficients with dotted and double dotted subscripts are damping and added masses coefficients, respectively. A set of linear equations with time-dependent coefficients is obtained when the variational equations are obtained. Defining the motions as:

z (t )  zˆ(t )   (t )  z a cos( e t   z )   (t )  (t )  ˆ(t )   (t )   cos( t   )   (t ) a

e



 (t )  ˆ(t )   (t )   a cos( e t    )   (t )

(4)

 (t ), (t ), (t ) are perturbations superimposed to the steady solutions ˆ ˆ zˆ, , , then, inserting expressions (4) into equations (1), (2) and (3), the

where

equations of motion relative to the perturbed motions become:

238 M.A.S. Neves et al.

1 1 1 (m  Zz )  Zz  Zθ  Zθ  Zz  Z  (Zzz zˆ  Zzˆ  Zzzzzˆ2  Zzˆ2  Zzˆ2  Zzz zˆˆ)  2 2 2 1 1 1 2 2 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ (Z  Zz zˆ  Z )  (Zz zˆ  Z  Zzz zˆ  Z  Z  Zz zˆˆ)  2 2 2 [Zz (t)  Z z (t)  2Z zz (t)zˆ  Z z (t)ˆ]  Zφ (t)  [Z (t)  Z (t)  Z z (t)zˆ  2Z (t)ˆ]  0

(5)

1 1 (Jx  Kφ )  K  Kˆ  K (Kzˆ  Kzz zˆˆ  Kzˆˆ) (Kz zˆ  Kˆ  Kˆ  Kzz zˆ2  2 2 1 Kθˆ 2  Kz zˆˆ ) (Kˆ  Kˆˆ  Kz zˆˆ)  2 Kz (t) [K (t)  K(t)  K z (t)zˆ  K (t)ˆ ]  Kθ (t)  0 (6) 1 1 1 (Jy  Mθ )  Mθ  Mz Mz  Mz Mθ (Mzz zˆ  Mzθθˆ  Mzzz zˆ2  Mzˆ2  Mzˆ 2 Mzz zˆˆ )  2 2 2 1 1 1 2 2 2 (M  Mz zˆˆ  M θˆ θˆ) (Mz zˆ M θθˆ  Mzz zˆ  Mˆ  Mˆ  Mz zˆˆ )  2 2 2 ˆ [Mz (t) M z (t) 2 M zz(t)zˆ  M z (t) ]  Mφ (t) [M (t) M (t) M z (t)zˆ 2M (tˆ) ]  0

(7) Equations (5 - 7) form a set of time-dependent coupled equations, assumed to regulate, to first approximation, the stability of the dynamical system defined by the non-linear equations defined previously, equations (1 - 3).

4 Roll Instabilization In the case of longitudinal waves, roll motions will not be externally excited, thus

ˆ  0 . The roll variational equation may then be expressed as:

1 1  ( J xx  K  )  K    K   K z zˆ  K  ˆ  K zz zˆ 2  K  ˆ 2  K z zˆˆ  K  t   K z t zˆ  2 2   K  t ˆ  K  t    0 

(8) In equation (8) the terms within brackets are restoring terms displaying the complexity of the parametric excitation, as determined by derivatives and the linear and squared vertical motions. The derivative coefficient K  corresponds to the linear roll restoring moment, that is, K   g . G M ; the derivative

An Investigation on Head-Sea Parametric Rolling for two Fishing Vessels 239

coefficients

K z , K , K zz , K and K z are associated with parametric

excitation due to nonlinear hydrostatic pressure terms, as shown in (Neves and Rodríguez 2006) and are related to derivatives of geometric characteristics of the waterplane area:

K z 

I xx I xx  2 I xx  2 I xx  2 I xx ; K   ; K zz  ; ; K  K   z z  z z 2  2

which, in turn, are dependent on the longitudinal distributions of breadth and flare of the hull. The following expressions have been derived (Neves and Rodríguez 2006):

I xx y  2 y 2 dx z z I y K   xx  2 xy 2 dx  z 2  2 I xx  y  K zz  4 y    z  dx  A0 z 2 K z 

2

K z

 2 I xx  y   4  xy  dx  A0 x f 0  z  z 

K 

 2 I xx  y   4  x 2 y  dx  I yy 0  2   z 

2

The oscillatory coefficients K

t , Kz t , K t 

(9) and

K t  describing

second and third order terms associated with parametric excitation due to nonlinear wave pressure terms are related to longitudinal distributions of breadth and flare of the hull and incident wave characteristics:

K  (t )  2g  y 2 L

y dx z

  y  2  K z (t )  g  4 y    2 y  dx   z  L   2    y  K  (t )  g  4 x y   2 x y dx  z   L 

240 M.A.S. Neves et al.

  y  2  K  (t )  g  2 y    y   2 dx   z   L 

(10)

where   x, t   Aw coskx  et  is the undisturbed wave elevation along the ship hull. More details of the mathematical model may be found in Neves and Rodríguez (2006). Equation (8) is a time-dependent equation describing parametric amplification of the roll motion (to third order), which contains contributions from the heave and pitch motions and the wave passage effect. Considering the harmonic character of the functions zˆ (t ),ˆ(t ) and Hill’s differential equation:

J

xx



 K   K 

K



 ( x, t ) , equation (8) typically corresponds to a



 a 2 R0  a R1C Coset  R1S Sinet   a 2 R2C Cos 2et  R2 S Sin 2et    0 (11)

It is pointed out that the time-varying terms are a compound result of volumetric changes in the submerged part of the hull due to the heave and pitch motions and the cyclic passage of the longitudinal progressive wave along the hull. It is well known that this type of equation may have strong instabilities at encounter frequencies close to ω e  2 ω n , where the roll natural frequency, squared, is defined as: 2

ωn 

Kφ J x  K φ

.

In the following sessions, the influence of speed on this problem will be discussed for the condition ω e  2 ω n and other conditions close to it. A relevant result at this stage is that the coefficients R 0 , R1S , R1C , R 2 S and R 2C which are all defined in terms of expressions (9, 10) define the intensity of the parametric excitation in equation (8) (or its equivalent equation (11)) are all dependent on the longitudinal distributions of sectional flare and breadth. These are analytical results that allow interesting conclusions with respect to the influence of hull form on the occurrence of parametric rolling, as will be discussed ahead. On the other hand, the intensity of the vertical motions are also relevant to the combined results of the terms appearing within brackets in equations (8) and (11). These will be examined in the next section, as they will be helpful for the objective of interpreting the experimental results.

An Investigation on Head-Sea Parametric Rolling for two Fishing Vessels 241

5 Vertical Motions Figures 3 to 6 present the computed heave and pitch linear responses (amplitude and phase) in the frequency domain for the two hulls. The tested frequencies are indicated by vertical dotted lines. It is observed that the heave responses for the RS and TS hulls, Figures 3 and 4, are practically the same, for zero speed and all other speeds tested. As expected, higher responses are obtained for higher speeds. RAO - Heave phase

RAO - Heave Amplitude Ship RS: ksi = 180°

1.20 1.00

Fn = 0.00 Fn = 0.10

135

Fn = 0.14 Fn = 0.20

90

0.80

Phase [deg]

Amplitude [m/m]

Ship RS: ksi = 180°

180

0.60 0.40

Fn = 0.00 Fn = 0.10 Fn = 0.14 Fn = 0.20

45 0 -45 -90

0.20 0.00 0.00

-135 0.50

1.00 1.50 2.00 We [rad/s]

2.50

-180 0.00

3.00

0.50

1.00 1.50 2.00 We [rad/s]

2.50

3.00

Fig. 3 Heave transfer functions, RS hull

For the pitch mode, Figures 5 and 6, some characteristics become apparent. First of all, one notices that for zero speed the two hulls respond in practically identical conditions. As speed increases, the responses become distinct. It may be observed that for the RS hull the pitch amplitudes have a general trend to increase with speed. In the case of the TS hull, there is a much more limited increase of the pitch amplitudes with speed. It is apparent, then, that if in the zero speed case the two hulls would show similar levels of internal excitation, as speed increases the RS hull would have to cope with higher levels of internal excitation due to the vertical motions. RAO - Heave Amplitude Ship TS: ksi= 180°

1.20 1.00

Fn = 0.00 Fn = 0.11 Fn = 0.15 Fn = 0.20

90

0.80 0.60 0.40

45 0 -45 -90

0.20 0.00 0.00

Ship TS : ksi= 180°

135

Fn = 0.15 Fn = 0.20

phase [deg]

Amplitude [m/m]

RAO - Heave phase

180

Fn = 0.00 Fn = 0.11

-135 0.50

1.00

1.50 2.00 We [rad/s]

2.50

3.00

-180 0.00

0.50

1.00

Fig. 4 Heave transfer functions, TS hull

1.50 2.00 We [rad/s]

2.50

3.00

242 M.A.S. Neves et al.

Yet, as will be made clear below, due to hull characteristics, this is not strictly the case. Another relevant aspect is that the influence of speed is effective not only in inducing a compounded effect on the amount of parametric excitation being transferred from the wave excited vertical motions to the rolling motion. It should also be taken into account that as different levels of internal excitation are fed into the dynamic system, there is also a higher level of damping incorporated into the process, its main mechanism being lift damping due to the higher speed. In fact, ship speed tends to (nonlinearly) reduce eddy damping and increase (linearly) lift damping (Himeno 1981). RAO - Pitch Amplitude

RAO - Pitch phase

Ship RS: ksi = 180°

12.00

315 270

8.00 Phase [deg]

Amplitude [deg/m]

10.00

Ship RS: ksi = 180°

360

Fn = 0.00 Fn = 0.10 Fn = 0.14 Fn = 0.20

6.00 4.00

225 180 135 90

2.00 45 0.00 0.00

0.50

1.00

1.50

2.00

2.50

0 0.00

3.00

Fn = 0.00 Fn = 0.10 Fn = 0.14 Fn = 0.20

0.50

1.00 1.50 We [rad/s]

We [rad/s]

2.00

2.50

3.00

Fig. 5 Pitch transfer functions, RS hull RAO - Pitch Amplitude Ship TS: ksi = 180°

12.00 10.00

Fn = 0.20

315

8.00

270

6.00 4.00

225 180 135 90

2.00

45 0.00 0.00

Ship TS: ksi = 180°

360

Phase [deg]

Amplitude [deg/m]

RAO - Pitch phase

Fn = 0.00 Fn = 0.11 Fn = 0.15

0.50

1.00 1.50 We [rad/s]

2.00

2.50

3.00

0 0.00

Fn = 0.00 Fn = 0.11 Fn = 0.15 Fn = 0.20

0.50

1.00 1.50 2.00 We [rad/s]

2.50

3.00

Fig. 6 Pitch transfer functions, TS hull

The net effect of speed on the balance of internal transfer of energy versus damping is not evident at all, and should be in any case carefully investigated. This observation leads to important considerations directly related to practical aspects of ship design for improved stability.

An Investigation on Head-Sea Parametric Rolling for two Fishing Vessels 243

At this stage, it is important to note that the RS hull is less damped than the TS hull. This was demonstrated by the roll decrement tests (conducted for the zero speed case). This may be observed in Figure 7, in which the following nondimensional forms are defined for roll damping and frequency of oscillation, B44

'

respectively: B44 

L   2

5

;   e

L g

.

Fig.7 Equivalent damping at zero speed

6 Results for RS Hull In graphical form, the experimental results for the RS hull given in Table II may be summarized as in Figure 8. The graphs show roll amplitude against wave amplitude at the tested Froude numbers and display all tested conditions. φ

40

φ

RS GMt = 0.34 m

40 RS GMt = 0.48 m

Fn = 0.10

30

Fn = 0.14 Fn = 0.20

30

Fn = 0.10 Fn = 0.14 Fn = 0.20

Fn = 0.34

20

20

10

10

0 0.20

0.40

0.60

a

0.80

1.00

1.20

0 0.20

0.40

0.60

a

0.80

1.00

1.20

Fig. 8 Variation of roll amplitude with wave amplitude for different Fn. RS hull: GM  0.34 m (left); GM  0.48 m (right)

In Figure 8 it is observed that for both values of GM and for each Fn there is in general a linear tendency in the growth of the roll amplitude for larger wave amplitudes. Additionally, this tendency is kept almost the same as larger speeds

244 M.A.S. Neves et al.

are considered. There is a clear tendency, when the same wave amplitude is considered, for roll amplitudes to become smaller at higher speeds. Figure 9 illustrates this tendency. The figure displays time series of roll motion obtained for the RS hull, GM  0.48m for three Froude numbers and a = 0.90m. Clearly, there is a marked reduction in roll amplification as speed increases. It is noted that for this value of metacentric height no roll amplification was observed for Fn  0.34 . 40

(a)

RS GMt=0.48 m Fn=0.10 ha=0.90 m

φ

20

0

-20

-40 0

20

40

60 t[s]

80

100

40

120 (b)

RS GMt=0.48 m Fn=0.14 ha=0.90 m

φ

20

0

-20

-40 0

20

40

60 t[s]

80

100

40 RS

120 (c)

GMt=0.48 m Fn=0.20 ha=0.90 m

φ

20

0

-20

-40 0

20

40

60

80

100

120

t[s]

Fig. 9 Time series of roll motion for RS hull, GM  0.48 m , a  0.90m , (a) Fn = 0.10; (b) Fn = 0.14 and (c) Fn = 0.20, respectively

The implication here is that with increased speed of advance, more damping (mainly lift damping) now acts against a higher level of vertical motions, but resulting in less roll parametric amplification. To help in the assessment of the balance between parametric excitation and damping, Figure 10 shows how internal inside the unstable zone of the stability diagram are the corresponding tested

An Investigation on Head-Sea Parametric Rolling for two Fishing Vessels 245

conditions for Fn = 0.10. The stability diagram is defined here in terms of encounter frequency versus wave height. GMt = 0.48 m

3.0

Fn=0.10

RS We=1.784 [rad/s]

2.5

hw/2 [ m ]

2.0

1.5

1.0

0.5

0.0 0.00

0.40

0.80

1.20

1.60

2.00

We [ r/s ]

Fig. 10 Stability diagram illustrating how internal the experimental points are for Fn = 0.10

Another way of viewing this roll reduction with speed is as shown in Figure 11. In this figure, final (steady) roll amplitudes are plotted against Froude number for one particular wave amplitude. For the lower metacentric height condition, Figure 11 (left), the tested wave amplitudes are not exactly the same for all speeds, thus linear interpolation has been adopted. But the important aspect to be observed from the two graphs is the common tendency for roll motion amplification to attenuate in face of higher speeds, irrespective of the frequency tuning. 40 40

RS GM=0.34m a=0.84m

RS GM=0.48m a=0.90m

30 Phi (deg.)

Phi (deg.)

30

20

10

0 0.00

20

10

0 0.10

Fn

0.20

0.30

0.00

0.10

0.20 Fn

0.30

Fig. 11 Roll amplitudes against Fn for RS hull: [ GM  0.34 m , a  0.84 m , e  1.49 rad / s ] (left); [ GM  0.48 m , a  0.90 m , e  1.78rad / s ](right)

246 M.A.S. Neves et al.

7 Results for TS Hull Consider now the results for TS hull given in Table III. These are shown in graphical form in Figure 12. It may be observed from Fig. 12 (left) that for the lower GM and small Froude numbers there is a tendency for roll angles to change in a similar way as discussed previously to the RS hull. Yet, for higher speeds, the tendency is reversed, and it is observed that very large roll angles are obtained for Fn=0.30.

30

φ

φ 40

TS GMt = 0.37 m

Fn = 0.11 Fn = 0.15 Fn = 0.20 Fn = 0.30

40

Fn = 0.11 Fn = 0.15

30

20

20

10

10

0 0.20

0.40

0.60

TS GMt = 0.50 m

a

0.80

1.00

1.20

0 0.20

0.40

0.60

a

0.80

1.00

1.20

Fig. 12 Variation of roll amplitude with wave amplitude for different Fn. TS hull, GM  0.37 m (left); GM  0.50 m (right)

It may be observed that for the lower GM and small Froude numbers there is a tendency for roll angles to change in a similar way as discussed previously for the RS hull. Yet, for higher speeds, the tendency is reversed, and it is observed that very large roll angles are obtained for Fn=0.30. In fact, considering the complete test program, the largest roll response was obtained for this condition, corresponding to angles of the order of 38 degrees being reached in few cycles, wave amplitude being a=0.78m. The time series corresponding to this impressive TS GMt = 0.37m Fn = 0.30 a = 0.78m

40

20

0

-20

-40 0

40

t[s]

80

120

Fig.13 Roll angle time series for TS hull at Fn=0.30, low GM, a=0.78m

An Investigation on Head-Sea Parametric Rolling for two Fishing Vessels 247

40

40

30

30 Phi (deg.)

Phi (deg.)

parametric roll amplification in head seas is given in Figure 13. This intense resonance for Fn=0.30 is larger than the roll amplitudes registered for zero speed, reported in (Pérez and Sanguinetti 1995, Neves et al. 1999). For zero speed, in less favourable conditions (GM=0.35m and a=0.90m), roll angles of 34 degrees were obtained. Still considering this low GM for TS testing, it is observed that at Fn=0.20 there are intense responses, though not as intense as in the Fn = 0.30 case. For the higher metacentric height, GM=0.50m, a tendency similar to what was observed for the RS hull is back: higher speeds now imply lower amplifications. It may be noticed that for TS hull at this high GM condition, for Fn>0.15 practically no amplification was observed. It may be observed that the situation for TS hull at the low GM condition is completely different from the RS hull responses in the high-speed range. Interpolating in Table III for the lower Froude numbers one gets Fig. 14 (left). When the same procedure is applied in the case of higher GM, for a=1.02m, one gets Figure 14 (right).

20

20

TS GM=0.37m a=0.78m

10

0 0.00

TS GM=0.50m a=1.02m

10

0.10

0.20 Fn

0.30

0 0.00

0.10

0.20 Fn

0.30

Fig. 14 Roll amplitudes against Fn for TS hull: [ GM  0.37 m , a  0.78 m , e  1.72 rad / s ] (left); [ GM  0.50 m , a  1.02 m , e  2.00 rad / s ](right)

It is seen in Figure 14 (left) that for TS hull in the range of low speeds corresponding to 0.11  Fn  0.15, roll amplitudes decay, as was the case with RS hull. But for higher speeds, instead of progressively lower responses, there is now an increase in roll response. In fact, responses are all high, but what is striking is the difference in trend. Now, it is seen that for the higher metacentric height GM=0.50m, Figure 14 (right), there is the same previous trend for roll amplitudes to decay with increased speeds. In fact, for Fn=0.20 and above this value, no amplification was observed for this condition.

248 M.A.S. Neves et al.

8 Summary of Comparisons One notices at first that the TS hull form is in general a more damped hull form than the RS hull, see Figure 7. Secondly, it is noticed that reports in general point out to less parametric amplification in head seas with speed (France et al. 2003). This is the observed trend in the present investigation for the RS hull in the two tested loading conditions. It is also the situation with the TS hull at the high metacentric height condition, GM=0.50m. What would be the explanation for the distinct trend for TS hull for the low GM condition? Given the coupling of the modes and the terms in the Hill’s equation, expressions (8) and equation (11), it may be concluded that the internal transfer of energy from the vertical modes to the roll motion is regulated essentially by hull form parameters such as the longitudinal distribution of local breadth b(x) and dy flare at waterline ( x) . dz 0 It is shown in Figure 15 that TS hull, due to its transom stern configuration displays much larger longitudinal asymmetry in flare distribution than the RS hull. Thus it is clearly a more efficient converter of energy from vertical modes to roll motion. In this context, it may be argued that for such critical dynamic characteristic to take place, metacentric height must be low. If this is not the case, then increased damping at high speeds will prevail against the unstabilizing effect of parametric excitation. In summary, low GM (GM=0.37m) for transom stern TS hull produces intense parametric resonances at all speeds at wave amplitudes of the order of a=0.78m. The other hull, which is less damped, even at a slightly lower metacentric height, GM=0.34m, and larger wave amplitude, a=0.90m, tends to respond less and less for higher speeds. The same TS hull for a high GM condition (GM=0.50m) does not amplify parametric excitation at high speeds, indicating that a transom stern shape together with a low GM has a relevant effect on parametric amplification. 5.00

y-coordinates at water line, RS hull dy/dz at water line, RS hull y-coordinates at water line, TS hull

breadth and flare at water line

4.00

dy/dz at water line, TS hull

3.00

2.00

1.00

0.00 -8.00

-4.00

0.00

Stations

4.00

8.00

12.00

Fig. 15 Longitudinal distribution of breadth and flare at waterline for the two hulls

An Investigation on Head-Sea Parametric Rolling for two Fishing Vessels 249

9 Roll Responses Around a Given Tuning Condition In the preceding sections the responses of roll motion in parametrically excited conditions have been examined. In all cases, for each metacentric height, as speed was modified, the ratio of the encounter frequency to the roll natural frequency remained the same, always corresponding to ω e  2ω n . In practice, it meant that in each case, for higher Froude numbers, longer waves excited the hull in the resonant condition. Different wave amplitudes allowed the analysis of the responses in a vertical sense in the stability diagrams, see Figure 9. In other words, the previous analysis is related to the investigation of speed effect on the tuning ωe  2ωn for a fixed metacentric height condition. It will be the next step to complementarily examine roll responses in parametric resonance when one single wave is considered (defining the tuning ωe  2ωn ) and distinct speeds are simulated, thus considering hull responses above and below one given tuning condition. we/ wn 1.25

1.50

1.75

2.00

2.50

RS GMt=0.48 m Wn=0.89 [rad/s] Fn=0.10

2.00

1.50

a [m] 1

2

3

4

5

6

1.00

0.50

0.00 1.20

1.60

2.00

we [rad/ s]

Fig. 16 Points corresponding to different speeds plotted in the stability diagram for Fn=0.10

In this sense, different speeds give the responses in a horizontal sense in the stability diagrams, see Figure 16. For the present purpose the Fn  0.10 condition will be examined. Thus, the tuning condition reported in Figure 8 (a) will be taken as reference for an assessment of the amplification of roll motion when ship speed is lower or higher than the one defining the exact tuning ωe  2ωn . Clearly, for

250 M.A.S. Neves et al.

the same wave, different speeds will define distinct values for the tuning ratio

ωe . ωn

Considering the limits of stability corresponding to Figure 9, defined in terms of wave amplitude against encounter frequency, points corresponding to different ship speeds in the same wave conditions (wave amplitude and length) will appear as indicated in Figure 16. Numbered points represent six different conditions (speeds) in head seas, in a wave with amplitude a  1.0 m and frequency ω  1.46 rad/s. For clarity, a second horizontal scale is displayed in the upper part of the figure, showing the ratio

ωe . One observes that point number 4 corresponds to the ωn

tuning ωe  2ωn . Time series obtained from numerical integration for constant conditions and varying speeds were analysed and the steady roll amplitudes (after transients) were plotted in a non-dimensional roll diagram in Figure 17, that is, Fn 

U gL

φ versus ka

. Therefore, roll amplitudes corresponding to the numbered points of

Figure 16 may be observed in Figure 17, plotted against a scale of Froude numbers. So, again, point number 4 corresponds to the Fn=0.10 condition, defining the exact tuning. It is seen that for very low speeds, corresponding to large detunings, the result is no roll amplification. Point 1 is in this category. Analogously, point 6 is a case of no roll amplification at the high-speed side. Maximum amplification in the response diagram takes place for Fn=0.07, between points 2 and 3. Roll amplification at the exact tuning is of the order of ¾ of the maximum amplification. There is a marked asymmetry in the response diagram. For speeds corresponding to Fn<0.065 (to the left of point 2) there is the appearance of a jump effect, with abrupt changes in roll amplitude, from large to very small roll amplitudes. On the other hand, the right hand side of the response curve displays a smooth reduction in amplitude as speed increases with corresponding larger detuning. Clearly, for all the other conditions related to the other tuning conditions tested, for instance, Fr=0.14, Fn=0.20, etc, similar curves will be obtained. Time series corresponding to conditions between points 1 and 2 effectively may display instabilities associated with jump phenomenon, as may occur in dynamic systems mathematically represented by the so-called externally excited Duffing equation. Yet, it is pointed out that the excitation in the present case is in effect an internal transfer of energy from the vertical modes to the roll mode, with speed effects as discussed in the previous session on the components of parametric excitation.

An Investigation on Head-Sea Parametric Rolling for two Fishing Vessels 251

RS We(Fn=0.10)=1.784 r/s GMt=0.48m

10 9

2

8

3

7 4

6

φ /ka

5 5

4 3 2 1 0 0.00

1

0.02

0.04

6

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

Fn

Fig. 17 Non-dimensional steady roll responses (

 ka

) for different Froude numbers

10 Conclusions Results of a series of experiments on parametric rolling undertaken for two fishing vessels in head seas have been presented. The first region of Mathieu instability was investigated. The parameters varied in the test programme were: wave amplitude, metacentric height and ship speed. Similar hulls tested in similar conditions displayed very distinct responses at specific testing conditions. The experiments demonstrate that in some cases strong parametric resonances in head seas can take place in quite few cycles. Angles of the order of 38 degrees have been reached for wave conditions often met by fishing vessels at sea. This is indicative that for this type of vessel, head seas conditions may be a source of real risk of ship capsize. A third order coupled non-linear mathematical model in the restoring modes was presented. The time-dependent linear equations defining the essential stability aspects of the nonlinear system were also given. The analytical aspects of the mathematical model were taken as a basis to explore different dynamical characteristics and explain distinct responses obtained in the experiments by the two hulls. The effect of speed on parametric resonance is strongly dependent on stern shape, but there seems to exist a complex interaction with other effects. A transom stern design, usually incorporating longitudinal asymmetry in the flare distribution, may exert a significant influence in establishing the tendency of a fishing vessel hull to display strong parametric amplification in head seas, particularly in a condition of low metacentric height. These conclusions are relevant in practice for hull design and operational considerations in rough seas. If a longitudinal distribution of flare is considered as

252 M.A.S. Neves et al.

a design solution, then proper account of the potentially higher parametric excitation should be given. It is pointed out that, as shown in the analysis, an increase in metacentric height is a possible via of solution to the problem, but this is not always available for some vessel types. This is clearly the situation with typical fishing vessels. Finally, considering steady roll responses around a given speed corresponding to the ωe  2ωn condition, it is observed that for speeds slightly lower than this reference speed, due to reduced damping and non-linear restoring characteristics of the hull, roll responses are larger than at the exact tuning. The response curve has a marked asymmetry with respect to the reference speed. Jump effect may be an associated dynamic feature in a roll amplification phenomenon due to parametric resonance.

Acknowledgement This investigation was partially supported by CNPq and FAPERJ of Brazil and CONICYT of Chile. The authors express their thanks for this financial support.

References Dallinga RP, Blok JJ, Luth HR (1998) Excessive rolling of cruise ships in head and following waves. RINA Int Conf on Ship Motions & Manoeuvrability. London. Feb France WN, Levadou M, Treakle TW, Paulling JR, Michel RK, Moore C (2001) An investigation of head-sea parametric rolling and its influence on container lashing systems. Mar Technol 2003 40:1 1-19 Himeno Y (1981) Prediction of ship roll damping - state of art. Dep of Nav Archit and Mar Eng. The Univ of Michigan 239 Kerwin JE (1955) Notes on rolling in longitudinal waves. Int Shipbuild Prog 16 Luth HR, Dallinga RP (1999) Prediction of excessive rolling of cruise vessels in head and following waves. Proc of PRADS Conf Neves MAS, Pérez NA, Valerio L (1999) Stability of small fishing vessels in longitudinal waves. Ocean Eng 26: 12 Neves MAS, Rodríguez CA (2006) On unstable ship motions resulting from strong non-linear coupling. Ocean Eng 33: 14-15 1853-1883 Paulling JR, Rosenberg RM (1959) On unstable ship motions resulting from nonlinear coupling. J of Ship Res 3: 1 Pérez NA, Sanguinetti CFO (1995) Experimental results of parametric resonance phenomenon of roll motion in longitudinal waves for small fishing vessels. Int Shipbuild Prog 42: 431 Sept

Simple Analytical Criteria for Parametric Rolling K.J. Spyrou School of Naval Architecture and Marine Engineering, National Technical University of Athens, 9 Iroon Polytechneiou, Zographos, Athens 157 80, Greece

Abstract Analytical design criteria are sought for predicting the parametric rolling behaviour of ships, taking into account: the growth rate during the inception stage of parametric rolling; and the steady amplitude that is reached when regular parametric excitation is assumed. The connection with the shape of the restoring curve and the effect of roll damping are shown. The results of the current deterministic treatment are transferred to the domain of probabilistic analysis by use of the concept of wave groups which seems to provide a powerful basis for criteria development.

1 Introduction The phenomenon of parametric rolling is known for at least half a century (Grim 1952; Kerwin 1995; Arndt & Roden 1958; Pauling & Rosenbersg 1959). Nonetheless, no specific design requirements referring to parametric rolling have found their way yet into the IMO stability regulations. A plausible explanation is that, whilst it is often the cause of intensive rolling, it is rarely documented to lurk behind a specific capsize accident. However, since the “nineties” the shipping world has become alert to this problem, realising that such exotic and in principle ‘non capsizal’ instabilities could incur tremendous effects in terms of loss or damage of property and business interruption (Gray 2001; Tinslay 2003). A classification society took the lead and published a guide for the parametric rolling of containerships (ABS 2004). By-and-large, the dynamics of the parametric rolling of ships should be considered in our days to be well understood (e.g Spyrou 2000, Neves 2002, Bulian et al 2003; Shin et al 2004). Nonetheless, some modern ship hull-forms, like post panamax containerships and probably some of the new large passenger ships, that are characterised by their heavily flared bow and flat stern with wide transom, may be sailing without having examined their tendency for parametric rolling in a longitudinal seaway. Besides issues of ship design and/or operation, currently, there is M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_13, © Springer Science+Business Media B.V. 2011

253

254 K.J. Spyrou

also discussion about the effectiveness of our physical model testing techniques for verifying that a hull form does not show tendency for parametric rolling (Belenky et al 2003). Whilst these issues should not be regarded as separate, here the focus will be set only on the question of design criteria. Specifically, we intend to put forward a new concept for assessing, at an early design stage, the tendency of a ship for parametric rolling, which combines the deterministic and probabilistic sides of the problem and encompasses the following three principles: Vulnerability:  The probability of exhibiting parametric rolling due to the encounter of a dangerous wave group should be kept lower than the acceptable level. Assessment of Post-Critical Behaviour: 

The amplitude of parametric rolling should not exceed a predefined, ship specific, limit of safe operation. Abrupt growth of roll within a small number of critical wave encounters should not be allowed.



We shall expand on these three principles, taking the last one first:

2 A Transient Criterion for Initial Parametric Roll Growth It can be shown that for a Mathieu system the unstable motion for exact resonance in the first region of instability builds-up according to the following approximate general law (Hayashi 1985):

 t   c1e  0 t sin  0 t     c 2 e   0 t sin  0 t   

where a 

4 02

 e2

(1)

(  0 is the natural roll frequency and  e is the wave encounter

frequency) and h

isthe scaled amplitude of GM variation. Also,  ,  are

functions of a , h determined, to first-order, from the expressions:

cos 2 

a 2 h 2  4a  12 2 a  1  (     0 ),    ah 4 2

(2)

Simple Analytical Criteria for Parametric Rolling 255

At a  1 the coefficient  obtains its maximum value  max   h 4 where    4 . From (1) and after substitution of the initial conditions  0    0 and  0   0 , we can extract the growth of amplitude after p roll cycles:

  p T0  e  0

p h 2

e 2



p h 2

(3)

In the presence of linear damping the growth rate is reduced:

 T0   wT0  e



h

2k

0



e

2

e 2



h 2



e





2k

0

4k  h 2  0

w0 

e

4k  h 2  0

   

e 2





4k  h 2  0

   

0

(4)

   

From the exponential term e we can extract the well-known condition of stability (that rules out the possibility of motion growth):

hcrit 

4k

(5)

0

The above is a condition of asymptotic stability; i.e. in principle a very long sequence of tuned waves having the ‘right’ height is required for instability. Apparently, there is an advantage in exploiting equation (4), which corresponds to transient rolling, rather than (5), which refers to asymptotic behaviour and leads to an unnecessarily more stringent requirement. On the basis of (4), roll growth at exact resonance, ( a  1 ) after p roll cycles, should be predicted from the following expression:

  pT0   e



2 pk

0

p h  p h  e 2  e 2  2  

   0  

(6)

256 K.J. Spyrou

A q-fold increase of roll amplitude from some initial angle value should entail p roll cycles:

ln q  

2p k

0

p h  p h  e 2  e 2  ln  2  

    

(7)

Since, after one or two roll cycles the exponential term of (7) with positive sign becomes dominant, the above may be written further, approximately, as:

ln q  

2p k

0



p h p  ln 2  2 2

 4k h    0 

   0.693  

(8)

It accrues that the number of cycles p required for a q-fold increase of amplitude may be obtained from the expression:

h

4k

0



0.693  ln q 1.571 p

(9)

The corresponding time t m should be p times the natural period T0 :

t q  p T0

(10)

To demonstrate the value of (9), let us think in terms of the following criterion: a 10-fold increase of roll amplitude should not come about in less than 4 roll cycles (which means 8 wave encounters – it is possible to link this to the probability of encountering a dangerous wave group). Let us for example consider waves with  L  1.0, H   1 20 . These translate into the following requirement for h, k :

h

4k

0



0.693  ln 10 3   0.477 1.57  4 6.282

(11)

In summary, by focusing on transient response, one can determine the critical parametric amplitude h , or equivalently the critical damping, for any wave group run length, thus achieving a meaningful interface with the probabilistic nature of

Simple Analytical Criteria for Parametric Rolling 257

ocean waves. The condition of asymptotic stability is recovered from (9) by setting p   . The criterion should be probably supplemented by a time requirement based on (10). For example, the 4 roll cycles should take 25.7  4  102.8 s. For very low natural frequencies, the required time becomes excessive. This leaves plenty of time for reaction (i.e. change of speed or heading) if the initiation of the phenomenon is promptly recognised.

3 Roll Growth Ends Nonlinearly on a Limit Cycle As is well known, there is no reason for this growth to persist up to infinity and thus lead by necessity to capsize. The detuning due to the nonlinear character of the GZ curve together with the increased dissipation due to the mild nonlinearity of damping, create the prospect of capture into steady oscillatory rolling with moderate amplitude. In effect, for a typical parametric growth with nonlinear restoring the boundary curves of stability discussed earlier represent loci of bifurcations giving birth to stable and unstable oscillatory behaviour (see for example Skalak & Yarymovych 1960; Soliman & Thompson 1992). The instability boundary curves of the upright state of a ship do not contain entirely the domain where parametric oscillations are realisable. The emerging stable roll oscillations need not be confined inside the “tongues” of the linear system and stable oscillations exist also well outside these regions (Scalak & Yarymovich 1961, Thompson & Soliman 1993). Should anyone worry about these nonlinear oscillations whose domain of existence extends outside the “tongues” of the linear parametric oscillator? The answer is, probably yes. In an idealised environment of a periodic seaway that is free from other external disturbances, a ship should find no dynamical reason for leaving the upright state as long as the combination of frequency ratio and parametric amplitude corresponds to some point in the region of stability. However, should the stable upright condition be sufficiently disturbed, this oscillatory behaviour can be incurred in an abrupt way. It may be said of course that, by moving away from the condition of exact resonance, the probability of occurrence of parametric rolling decreases. However, it is doubted whether it is safe to assume that this probability is sufficiently low so that the phenomenon can be completely ignored. Perhaps one should place less emphasis on the necessity of fulfilling the condition of exact principal resonance for the occurrence of parametric rolling. To explain these points further, let us consider a Mathieu-type roll equation with a single, cubic nonlinear term:

258 K.J. Spyrou

  2k    02 1  h cos e t   n 02  3  0

(12)

The constant n could be negative, in which case we are examining the oscillations corresponding to the initial part of the GZ   curve which is often of “hardening spring” type. Or we could consider it as positive, in which case we refer generically to the entire GZ   curve up to the angle of vanishing stability. Application of a perturbation method like harmonic balance or averaging leads to the following explicit formula for the amplitude:

A2 

4  1  1    3 n  a  

h 2 4k 2  4 a 02

   

(13)

Setting A  0 we find the curve whereon the oscillations are created. It comes to no surprise that this curve is independent of the nonlinear coefficient n and it coincides with the boundary of linear stability. Also, the term inside the square root, as well as the whole expression of A 2 , should be non-negative. For an initially hardening restoring ( n  0 ) these yield,

h

4k

0 a

and a  1

(14)

In essence, (14) defines the locus of “saddle-node” (or so-called “fold”) bifurcations. On this curve, the unstable periodic states that are shed from the left boundary of the instability region undergo a U turn and become stable (this can be understood by considering that, for negative n , at the ‘lower a ’ boundary of the instability region a subcritical bifurcation takes place, creating unstable oscillations; whereas the boundary at a  1 is a supercritical one i.e. stable oscillations are created). Condition (14) determines the true boundary of periodic response. For a certain level of h , the region where stable oscillations are encountered is wider than the predicted from the linear analysis. In Fig. 1 is seen the development of steady roll 4 amplitude as function of a and h ( A  A *  ) for a container with the 3n dimensions of APL China and  0  0.2448 s 1 , k  0.015 s . The stability of the emerging steady roll oscillations is indicated (dashed line represents unstable states).

Simple Analytical Criteria for Parametric Rolling 259

A* 0.8 0.8 0.9

0.6

1.1 1.2

0.4 0.2 0 0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 h

Fig. 1 Amplitude of response for hardening restoring

The domain of oscillatory behaviour (bounded by the thick continuous line) can easily be found with some manipulation: 2

 3n 16k 2  1  h  4  A 2  1     2 0 a  a  4

(15)

The combinations of h, a  that give rise to oscillations of predefined A* is shown in Fig. 2. It is not difficult to prove that the descending part of each iso- A* curve corresponds to stable rolling and the ascending to unstable. The boundary of stable rolling is reconfirmed. As we have multiple coexisting stable responses, the initial conditions and the availability of sufficiently strong external disturbances determine whether the ship can stay upright; or she should adopt the one (desired) or the other (undesired and possibly dangerous) way of behaviour. In Fig. 3 is shown the variation of the roll amplitude A for n=-2.35, as a function of the linear damping and the amplitude of parametric forcing. For small (yet realistic for several operating ships) damping, the effect on the response amplitude is relatively small. The same applies for h. A more influential parameter is the coefficient of the cubic stiffness term n which, at first approximation, is linear to the response amplitude.

260 K.J. Spyrou

1

0.7

h

0.6 0.8

0.5

0.6

0.4

0.4

0.3 0.2 0.1 h = 4k/(w0 a) 0.6

0.7

0.8

0.9

1

1.1

a

Fig. 2 Iso-A curves (from 0.1 to 0.7) for k=0.015, n=-2.35 (hardening),

1.2

 0 = 0.2448 s-1

A

/

1 2

0.8

x

1x

0.6 2x 0.4 3x 0.2 0 0

0.01

0.02

0.03

0.04

k

0.05

Fig. 3 Effect of damping on the amplitude of periodic response with parameter the coefficient of nonlinear stiffness (“hardening”)

3 Effect of the Fifth Order Term (Initially Hardening, then Softening): Consider again the roll equation, this time with a fifth-order polynomial for restoring which can take better into account the details of the GZ curve up to large inclinations:

Simple Analytical Criteria for Parametric Rolling 261

d 2 d

2



2 k a d  a 1  h cos 2   na  3  ma 5  0  0 d

(16)

The approximate steady-state solution is:

A2  

2 1 3 n 8  3 n  1       5m 5m a 5 m 

h 2 4k 2  4  02 a

   

(17)

We select a GZ curve (see Fig. 4) very close to that of the post-panamax containership of France et al (2001). The selected values for the coefficients n and m are, respectively, -0.14 and 0.25. The amplitudes as functions of the parametric term h, for the frequency ratios examined earlier, are shown in Fig. 5. Several changes of stability are noted on each one of these curves. An interesting feature of this diagram is that it shows the behaviour at large angles where the fifth order term of the restoring function becomes influential. Contrast of Fig. 5 versus Fig. 1 is enlightening in this respect.

4 Requirement of Limited Amplitude of Steady Parametric Rolling A supplementary criterion based on the steady roll amplitude may thus be introduced at this stage, as a second requirement concerning post-critical behaviour: For a critical wave, say with  L  1.0, H   1 20 , the max roll amplitude should not exceed, say, 15 deg (this value is proposed by ABS; perhaps this value can be vessel specific depending e.g. on lashings’ strength). The combination of restoring and damping coefficients that satisfy the requirement of no exceedance of this limiting angle can be obtained with some manipulation of (13); or of (17) if judged that the later part of the GZ curve should also be considered (for a 15 deg limit it is unlikely this to be necessary).

262 K.J. Spyrou GZ(j) 1.5 1.25 1 0.75 0.5 0.25

0.2

0.6

0.4

0.8

j

1

Fig. 4 The exact (dots) and the polynomial fit (line) of the considered GZ curve.

A 1 0.8 0.6

1.1

0.4

1.2 0.9

0.2 0.8 0 0.3

0.4

0.5

0.6

0.7

0.8

h Fig. 5 Amplitude of roll oscillation for 5th order restoring, assuming time-dependence only for the linear term. The different curves are according to the value of the frequency parameter a.

5 Interfacing with the Probabilistic Seaway: The Key Is in the Groupiness of High Waves It is well known that higher waves tend to arise in groups. As the nearly regular characteristics of waves in a group are essential for giving rise to fundamentally resonant motions like parametric rolling, there is a meaningful link between the

Simple Analytical Criteria for Parametric Rolling 263

probabilistic nature of ocean weaves and the deterministic analysis. The probability of occurrence of parametric rolling could be assumed to be equal to the probability of encountering longitudinally a wave group with sufficient run length and exceeding the threshold height (determined from the deterministic analysis), given that the frequency falls in the critical range (which however has to be very wide). One may use theoretical or parametric models for joint distributions of wave parameters. In practical terms one has to make certain assumptions about the correlation of key parameters in a wave group and opt to use available bivariate distributions either of wave height and period, or of successive wave periods, or finally of successive wave heights. Ocean wave statistics suggest that height H and period T are correlated (e.g. Longuet-Higgins 1975). The following bivariate distribution proposed by Longuetm m  m2 Higgins (1983) is based on the spectral width parameter   0 2 2 1 (which m1 has the advantage to depend only on the first three moments):



p H , T  

 f    H *  4

where H * 

 H*

2

  e  T* 

H H

, T* 

4

2

2   1 1  T*  1  2  

T , Tz

f   

      

(18)



2 1  2







is the spectral width

1  2 parameter, m 0 , m1 , m 2 are respectively zeroth, first and second moment of the wave spectrum, H is the mean wave height; and Tz is the mean zero-upcrossing period that is determined from spectral moments, T z  2

m0 . m2

The use of joint distributions of successive wave periods for assessing probabilistically resonant ship rolling in beam seas was investigated by Myrhaug et al. (2000). However, unlike parametric rolling, the wave frequency where resonance occurs is well defined in a beam sea because the speed of the ship does not influence the encounter frequency. Moreover, to be initiated, it does not require a threshold wave height, like in the case of parametric rolling. Hence, an approach based on the statistics of successive wave heights, i.e. the condition of having a wave group with heights exceeding a known critical level, seems to be more relevant for the investigation of parametric rolling. The probability to encounter a sequence of waves with height above the critical level H c was considered by Blocki (1980) using the approach of Goda (1976). He

264 K.J. Spyrou

had assumed the occurrence of successive heights above H c as independent events. This means that practically, the probability of encountering a certain run length was underestimated. The degree of correlation of successive wave weights depends on the sharpness of the spectral peak. For the effect of the spectral bandwidth on the distribution of wave height see for example Kimura (1980), Tayfun (1983) and Longuet-Higgins (1984). Stansell et al (2002) found that, as bandwidth increases, there is a rather slight reduction in the mean run and group length, up to a bandwidth   0.6 beyond which they become rather insensitive (to obtain a sense of magnitude we note that   0.425 for a Pierson Moskowitz and   0.389 for a JONSWAP spectrum). According to Tayfun, the sharpness of the spectral peak reflects the variability of height between successive waves; and spectral peakedness is best represented by the correlation coefficient of the wave envelope R HH which could be calculated as follows:

R HH 



E    1   2 1

 K2   4



4



4 6    2      16  4  16 64   



(19)

E  , K   are complete elliptic integrals, respectively of the first and second kind. The correlation parameter  could be calculated as follows (see Stansell et al. 2002 for an extensive discussion on alternative methods of calculation):

 T  

1 m0

A2  B 2 , A 





0

E  f  cos 2f T df , B 





0

E  f  sin 2f T df (20)

Goda (1976) has found that for swells the correlation coefficient R HH is about 0.6 while for wind waves it is only about 0.2. Assuming that successive wave heights follow a Rayleigh distribution, Kimura (1980) derived the following bivariate probability density function pH 1 , H 2  for consecutive wave heights:

p HH H 1 , H 2  



4H 1 H 2

1   H 2

2 rms

e

H  H  1 H 2 1 2

2 2 2 rms

 2  H1 H 2 I0  1 2 H 2 rms 





   

(21)

where H rms is the root mean square wave height and I 0 is the modified Bessel function of zeroth order. The probability of having two consecutive wave heights above the critical height H c will be then:

Simple Analytical Criteria for Parametric Rolling 265

PH i 1  H c H i  H c  =



 Hc





p HH H 1 , H 2  dH 1 dH 2

Hc





Hc

p H H  dH

(22)

where p H H  is the marginal probability density which is Rayleigh type:

p H H  

2H 2 H rms



e

H2 2 H rms

(23)

The assumption of Markov chain for successive wave heights leads to the following probability function for the occurrence of a group with length j and peaks higher than H c which is in fact the probability of occurrence of parametric rolling:

Ppr  P j 1 1  P 

(24)

The above value should be multiplied by a factor indicating whether the speed range of the ship produces encounter frequencies that overlap with the frequencies of principal resonance.

6 Concluding Remarks The transient growth of parametric roll could be regarded as the result of a ship’s encounter with some wave group having suitable characteristics to incite such behaviour in a realistic seaway. Then, the probability of occurrence of parametric roll could be obtained from the probability of encountering longitudinally a wave group with sufficient run length and exceeding the threshold wave height, as determined from the deterministic analysis, given that the frequency falls in the critical range. One may exploit the well developed literature of theoretical or parametric models for the joint distributions of wave parameters of a Gaussian sea. Such a framework seems to be physically meaningful enough and practical for criteria development.

References ABS 2004 Guide for the parametric roll for the design of container carriers (draft), Houston, USA, 40 pages. Arndt B and Roden, S. 1958 Stabilität bei vor- und achterlichen Seegang. Schiffstechnik, 5 (29), 192-199.

266 K.J. Spyrou Belenky VL, Weems KM, Lin WM, Paulling JR (2003) Probabilistic analysis of roll parametric resonance in head seas. Proc, 8th Int Conf on the Stab of Ships and Ocean Veh, Madrid, 325340. Bulian G, Francescutto A and Lugni C (2003) On the nonlinear modeling of parametric rolling in regular and irregular waves, Proc, 8th Int Conf, Stab of Ships and Ocean Veh, 305-324, Madrid. Blocki W (1980) Ship safety in connection with parametric resonance of the roll. Int Shipbuilding Prog, 27, 36-53. Esparza I and Falzarano JM (1993) Nonlinear rolling motion of a statically biased ship under the effect of external and parametric excitations. Proc, Symposium on Dyn and Vibration of Time-Varying Systems, OE, 56, ASME, 111-122. Francescutto A and Dessi (2001) Some remarks on the excitation of parametric rolling in nonlinear modeling. Proc, 5th Int Workshop, Stab and Oper Safety of Ships, Trieste, 4.9.1-4.9.8. Goda Y (1976) On wave groups. Proc, BOSS’76, Vol. 1, 115-128. Gray M (2001) Rolling case for more research, Lloyd’s List (Section: Insight and Opinion), February 18, London. Grim O (1952) Rollschwingungen, Stabilität und Sicherheit im Seegang. Schiffstechnik, 1 (1), 10-21. Hayashi C (1985) Nonlinear Oscillations in Physical Systems, Princeton Univ Press, ISBN 0-691-08383-5, Princeton, New Jersey. Kerwin JE (1955) Notes on rolling in longitudinal waves. Int Shipbuilding Prog, 2 (16) 597-614. Kimura A (1980) Statistical properties of random wave groups. Proc, 1st Coastal Eng Conf, Vol. 2, 2955-2973. Longuet-Higgins MS (1975) On the joint distribution of the periods and amplitudes of sea waves. J of Geophys Res, 80, 6778-6789. Longuet-Higgins MS (1983) On the joint distribution of wave period and amplitudes in a random wave field. Proc of the Royal Soc, London, Series A, 310, 219-250. Longuet-Higgins MS (1984) Statistical properties of wave groups in a random sea state. Proc of the Royal Soc, London, Series A, 389, 241-258. Myrhaug D, Dahle EA, Slaattelid OH (2000) Statistics of successive wave periods with application to rolling of ships. Int Shipbuilding Prog, 47, 451, 253-266. Neves M (2002) On the excitation of combination modes associated with parametric resonance in waves. Proc, 6th Int Ship Stab Workshop, Webb Inst, Long Island. Paulling JR and Rosenberg RM (1959) On unstable ship motions resulting from nonlinear coupling. J of Ship Res, 3, 1, 36-46. Shin YS, Belenky VL, Paulling JR, Weems KM, Lin WM (2004) Criteria for parametric rolling of large containerships in longitudinal seas, SNAME Annual Meeting, Sept 30 – October 1, Washington D.C. (early copy). Skalak R and Yarymovych MI (1960) Subharmonic oscillations of a pendulum. J of Appl Mech, 27, 159-164. Soliman M and Thompson JMT (1992) Indeterminate sub-critical bifurcations in parametric resonance. Proc of the Royal Soc of London, Series A, 438, 433-615. Spyrou KJ (2000) Designing against parametric instability in following seas, Ocean Eng, 27, 625-653. Stansell P, Wolfram J and Linfoot B (2002) Statistics of wave groups measured in the northern North Sea: comparisons between time series and spectral predictions. Appl Ocean Res, 24, 91-106. Tayfun A (1983) Effects of spectrum bandwidth on the distribution of wave heights and periods. Ocean Eng, 10, 107-118. Tinsley, D. 2003 DNV Project thinks about the box. Lloyd’s List (Section: Shipbuilding and Shiprepair) . November 18, London.

Experimental Study on Parametric Roll of a Post-Panamax Containership in Short-Crested Irregular Waves Hirotada Hashimoto*, Naoya Umeda* and Akihiko Matsuda** *Osaka University, 2-1, Yamadaoka, Suita, Osaka, 565-0871, Japan **National Research Institute of Fisheries Engineering,7620-7, Hasaki, Kamisu, Ibaraki, 314-0408, Japan

Abstract Free running model experiments of a post-Panamax containership were conducted to examine the danger of parametric roll in regular waves, long-crested and shortcrested irregular waves. As a result, parametric roll in head and bow waves were clearly recorded even in long-crested and short-crested irregular waves. From the experimental results, the effects of wave steepness, forward velocity, heading angle, irregularity of waves on parametric roll were systematically examined.

1 Introduction Recent accidents of a post-Panamax containership due to parametric roll in head waves (France et al, 2003) forced the International Maritime Organization (IMO) to start to revise the Intact Stability Code (2002) and the guidance for the master in following and quartering waves (MSC Circ. 707, 1995). Although several experimental data were published for parametric roll even in irregular waves (Umeda et al, 1995), only limited outcomes are available for that in head and bow waves. In this revision work at the IMO, the following questions were now raised to be urgently solved; does the parametric roll in head waves lead to not only the cargo damage but also ship capsize, does the danger of parametric roll decrease in irregular waves, especially in short-crested waves, in comparison with an ideal regular waves, and how can we avoid and predict parametric roll in realistic head or bow waves? Responding to this situation, a free running model experiment mainly focusing on irregular head and bow waves was conducted to directly obtain the answers for the above questions. From the experimental results, the effect of wave height on parametric roll in regular waves, the effects of heading angle and forward velocity on parametric roll in regular waves, the effect of irregularity of waves on parametric roll in head waves, the effects of heading angle and forward velocity on parametric roll in short-crested irregular waves were systematically examined. M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_14, © Springer Science+Business Media B.V. 2011

267

268 H. Hashimoto et al.

2 Free Running Model Experiment 2.1 Outline of Model Experiments A free running model experiment was conducted in the basin of National Research Institute of Fisheries Engineering (length: 60m, width: 25m, depth: 3.2m) with an 80-segmented wave maker. A scaled ship model of a post-Panamax containership was used, and her principal particulars and body plan are shown in Table.1 and Figure 1, respectively. The ship model had no deck house, but was watertight, and was propelled by an electric motor with a constant revolution control system. An onboard computer realised an autopilot system of the constant gain of 1.0 and stored data of roll, pitch and yaw motions obtained by a fiber-optic gyroscope. In addition, surge and sway velocity were obtained from data of ship position recorded by an optical tracking sensor fixed to the basin. In regular wave runs, several sets of propeller revolution number, nc, wave height, H, wave length, , autopilot course, c, were used but the wave length to ship length ratio was fixed to be 1.6. Furthermore vertical displacement of incident waves at the centre of ship gravity was obtained in consideration of wave phase velocity, ship position and wave height records. In long-crested irregular wave runs, significant wave height H1/3 of 0.221m in model scale and mean period T01 of 1.3s, which correspond to the wave height and period of regular waves, were used with the ITTC spectrum. In short-crested wave runs, H1/3 and T01 were the same as the long-crested irregular wave runs, but with cosine to the 2nd or 4th power as the directional distributions function were used. By utilising the single summation method for generating wave signals, uniformity of short-crested waves Table 1. Principal particulars of the subject ship Item length between perpendiculars: L

Ship

Model

283.8m

2.838m

breadth: B

42.8m

0.428m

depth: D

24.0m

0.24m

draught at FP: Tf

14.0m

0.14m

mean draught: T

14.0m

0.14m

draught at AP: Ta

14.0m

0.14m

block coefficient: Cb

0.630

0.630

longitudinal position of centre of

5.74m

0.0574m

pitch radius of gyration: Kyy/Lpp

0.258

gravity from the midship: XCG

aft

Aft

metacentric height: GM

1.06m

0.0106m

natural roll period: T

30.3s

natural pitch period: T

3.20s 0.86s

Experimental Study on Parametric Roll of a Post-Panamax Containership 269

in space is realised within 5% error (Sera and Umeda, 2000). In case of the irregular wave, model runs were repeated so that number of encounter waves is about 150 in average. A photograph of the model runs in short-crested irregular waves is shown in Figure 2 as an example.

Fig. 1 Body plan of the subject ship

Fig. 2 A photograph of the free running model experiment in short-crested head waves

2.2 Experimental Results Examples of measured time history of head sea parametric roll in regular waves, long-crested irregular waves and short-crested irregular waves are shown in Figs. 3-5. Here propeller revolution number is the same and H1/3 and T01 are corresponding to the wave height and period of regular waves. In the Fig.3, parametric roll with its steady amplitude of about 18 degrees can be clearly recorded after 10 seconds. In long-crested irregular waves, parametric roll was also observed with the maximum roll angle of about 20 degrees as shown in Fig.4. Even in short-crested waves shown in Fig. 5, parametric roll does not disappear and the maximum roll angle of 22 degrees can be found. These results demonstrate that wave irregularities or short-crestedness does not exclude danger of parametric roll in head waves.

270 H. Hashimoto et al. 30

angle (degrees)

20 10 0 0

10

20

30

40

50

60

roll pitch

-10 -20 -30

time (s)

Fig. 3 Time history of parametric roll in regular waves (H/=1/20.6, /L=1.6, c=180 degrees and nc=15rps) 30

angle (degrees)

20 10 0 0

20

40

60

80

100

120

roll pitch

-10 -20 -30

time (s)

Fig. 4 Time history of parametric roll in long-crested irregular waves (H1/3=0.221m, =1.32s, c=180 degrees and nc=15rps) 30

angle (degrees)

20 10 0 0

20

40

60

80

100

120

roll pitch

-10 -20 -30

time (s)

Fig. 5 Time history of parametric roll in short-crested irregular waves (H1/3=0.221m, =1.32s, cos2 distribution, c=180 degrees and nc=15rps)

Experimental Study on Parametric Roll of a Post-Panamax Containership 271

The effect of wave height on parametric roll in regular head waves is shown in Fig. 6. Here, wave length to ship length ratio is 1.6 and propeller revolution number was set to realise ship forward velocity satisfying the ratio of 2.0 between encounter frequency and roll natural frequency. Parametric roll can be found even in small wave steepness. With increase of wave steepness, parametric roll with its amplitude around 20 degrees was observed at H/=0.0176, and the amplitude does not significantly decrease up to H/=0.0528. Finally parametric roll disappears at H/=0.0726. This can be explained that a ship condition deviates from a parametric roll condition with change in mean of restoring variation. Because H/ of 0.072 corresponds to the wave height of 32m in full scale and the maximum roll angle remains about 23 degrees, it is difficult to find direct relationship between the observed parametric roll and capsizing. The effects of heading angle and forward velocity on parametric roll in regular waves are shown in Fig. 7. Here, the heading angle of 180 degrees means a pure head sea condition, and 270 degrees does a beam sea condition. The Froude number is estimated with the speed loss due to wave taken into account. In head waves, several runs of parametric roll with its maximum roll angle of about 20 degrees were found in smaller Froude number. At the Froude number of 0.1, parametric roll suddenly disappears. When the heading angle becomes larger the maximum roll angle is smaller than the case of 180 degrees of heading angle, but parametric roll itself does not disappear even in relatively high forward velocity while it does in head waves. This is because the area that satisfies the parametric roll condition shifts to higher forward velocity with increase of heading angle. However, if we focus on severe parametric roll whose maximum angle is larger than 15 degrees, the danger decreases with increase of heading angle. The effects of irregularity of incident waves on parametric roll in head waves are shown in Fig. 8. In cases of smaller forward velocity, roll angles both in longcrested and short-crested irregular waves are slightly smaller than that in regular waves. Here some statistical fluctuation due to non-ergodicity (Belenky, 2004) can be found. When the Froude number is larger than 0.1, however, parametric roll in irregular waves still occurs while no parametric roll occurs in regular waves. This tendency is more conspicuous in the case of short-crested irregular waves. The effect of heading angle on the maximum roll angle of parametric roll in short-crested irregular waves is shown in Fig. 9. Here cosine to the 2nd power was used as the directional distribution function. This figure shows that the ship cannot avoid parametric roll only by changing its course in short-crested irregular waves, and also shows that increasing the speed is effective to reduce danger of parametric roll at least in the heading angle ranging from 180 degrees to 240 degrees.

272 H. Hashimoto et al.

amplitude of parametric roll (deg.)

25 20 15 10 5 0 0

0.02

0.04

0.06

0.08

wave steepness

Fig. 6 Effect of wave height on parametric roll in regular waves (/L=1.6 and c=180 degrees) amplitude of parametric roll (deg.)

25 180 deg. 203 deg.

20

225 deg. 15

248 deg. 270 deg.

10 5 0 0

0.05

0.1

0.15

0.2

0.25

0.3

Froude number

Fig. 7 Effect of heading angle and forward velocity on parametric roll in regular waves (H/=1/20.6 and /L=1.6)

amplitude of parametric roll (deg.)

25

regular waves long-crested waves

20

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

15 10 5 0 0

0.05

0.1

0.15 0.2 Froude number

0.25

0.3

0.35

Fig. 8 Effect of irregularity of waves on parametric roll in head waves (H1/3=0.221m, =1.32s and c=180 degrees)

Experimental Study on Parametric Roll of a Post-Panamax Containership 273

amplitude of parametric roll (deg.)

25 180 deg. 210 deg.

20

240 deg. 15 10 5 0 0

0.05

0.1

0.15 0.2 Froude number

0.25

0.3

Fig. 9 Effect of heading angle on parametric roll in short-crested irregular waves (H1/3=0.221m, =1.32s and cos 2  distribution)

2.3 Statistical Analysis Statistical analysis focusing on instantaneous angles and amplitudes of roll and pitch was carried out with recorded time histories. Probability density functions were obtained for both instantaneous angles and amplitudes of roll and pitch, and compared them with the Gaussian distribution and the Rayleigh distribution, respectively. Examples of the results in long-crested irregular waves are shown in Figs. 1011. Here the experimental condition is H1/3=0.221m, T01=1.32s, and autopilot course c=180 degrees. As shown in Figs. 10-11, the instantaneous pitch angle agrees with the Gaussian distribution and the pitch amplitude agrees with the Rayleigh distribution. Thus a linear theory can explain experimental results of the pitch motion even with quite high wave height of 22.1m in full-scale. On the other hand, probability density functions of instantaneous angle and amplitude of roll around 0 degrees are conspicuously large, and does not follow the Gaussian and Rayleigh distributions at all. These results clearly indicate that parametric roll is a strongly nonlinear phenomenon. Pitch motion immediately starts whenever a ship meets incident waves while a ship starts to roll only when a ship meets the wave group that exceeds a parametric roll threshold. Therefore the probability density around 0 degrees becomes large. For this reason, a conventional seakeeping theory based on a linear or weakly nonlinear assumption cannot be applied to parametric roll problems. Similar results have been reported by Belenky et al. (2003) and Ribeiro e Silva et al. (2005) based on their numerical simulations, however, no publication based on model experiments can be found so far. Examples of the results in short-crested irregular waves are shown in Figs 12-13. Here the experimental condition is H1/3=0.221m, T01=1.32s, autopilot course c=180

274 H. Hashimoto et al.

0.16

0.16

0.14

0.14

0.12

0.12

probability density

probability density

degrees and cos2  distribution. The instantaneous pitch and the pitch amplitude agree with the Gaussian and the Rayleigh distributions, respectively. Instantaneous roll angle and roll amplitude do not agree with each distribution like long-crested irregular wave runs, however, its difference becomes smaller to some extent.

0.1 0.08 0.06 0.04 0.02 0 -30

0.1 0.08 0.06 0.04 0.02

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

0 -20

30

-10 0 10 elevation of pitch angle (degrees)

20

Fig. 10 Probability density function of instantaneous roll and pitch angle in long-crested irregular waves (histogram: experiment; dashed line: Gaussian distribution) 0.25 0.2

0.15

probability density

probability density

0.2

0.1

0.05

0 0

0.15 0.1 0.05

5 10 15 20 25 amplitude of roll angle (degrees)

0 0

30

5 10 15 amplitude of pitch angle (degrees)

20

0.16

0.16

0.14

0.14

0.12

0.12

probability density

probability density

Fig. 11 Probability density function of roll and pitch amplitude in long-crested irregular waves (histogram: experiment; dashed line: Rayleigh distribution)

0.1 0.08 0.06 0.04

0.06 0.04 0.02

0.02 0 -30

0.1 0.08

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

30

0 -20

-10 0 10 elevation of pitch angle (degrees)

20

Fig. 12 Probability density function of instantaneous roll and pitch angle in short-crested irregular waves (histogram: experiment; dashed line: Gaussian distribution)

Experimental Study on Parametric Roll of a Post-Panamax Containership 275 0.2

0.3 0.25 probability density

probability density

0.15

0.1

0.05

0.2 0.15 0.1 0.05

0 0

5 10 15 20 25 amplitude of roll angle (degrees)

30

0 0

5 10 15 amplitude of pitch angle (degrees)

20

Fig. 13 Probability density function of roll and pitch amplitude in short-crested irregular waves (histogram: experiment; dashed line: Rayleigh distribution)

3 CONCLUSIONS Free running model experiments of a post-Panamax containership were systematically conducted to directly examine the danger of parametric roll in head and bow waves in regular waves, long-crested and short-crested irregular waves. As a result, the following conclusions are obtained: 1. 2. 3. 4. 5. 6. 7.

Capsizing due to parametric roll in head or bow waves was not observed for the post-Panamax containership model with the designed GM value. Maximum roll amplitude of parametric roll does not always increase with wave steepness in regular waves. It is recommended to avoid ±45degrees of heading angle from head sea condition to prevent severe parametric roll exceeding 15 degrees of roll amplitude in regular waves. Maximum roll angle of parametric roll in long-crested irregular waves or shortcrested irregular waves is almost the same as large as the steady amplitude in regular waves. Maximum roll angle of parametric roll does not depend on heading angle in short-crested irregular bow waves so much. Increasing ship forward velocity could be recommended to reduce the parametric roll danger in irregular head and bow waves, as an alternative to decreasing it. Instantaneous roll angle and amplitude of parametric roll cannot be explained as the Gaussian and Rayleigh distributions, respectively.

Acknowledgements This work was supported by a Grant-in Aid for Scientific Research of the Japan Society for Promotion of Science (No. 15360465). The work was partly carried out as a research activity of the RR-SP4 research panel of the Shipbuilding Research

276 H. Hashimoto et al.

Association of Japan in the fiscal year of 2004, funded by the Nippon Foundation. The authors express their sincere gratitude to the above organisations.

References Belenky VL (2004) On Risk Evaluation at Extreme Waves. Proceedings of the 7th Int Ship Stab Workshop 188-202 Belenky VL, Weems KM, Lin WM et al (2003) Probabilistic Analysis of Roll Parametric Resonance in Head waves. Proc of the 8th Int Conf on Stab of Ships and Ocean Veh 325-340 France WL, Levadou M, Treakle TW et al (2003) An investigation of head-sea parametric roll and its influence on container lashing system. Marine Technol 40(1) 1-19 IMO (2002) Code on Intact Stability for All Types of Ships Covered by IMO Instruments IMO (1995) Guidance to the Master for Avoiding Dangerous Situations in Following and Quartering Waves. MSC Circular 707 Ribeiro e Silva S, Santos TA, Guedes Soares C (2005) Parametrically Excited Roll in Regular and Irregular Head waves. Int Shipbuilding Prog 52(1) 29-56 Sera W, Umeda N (2000) Homogeneity of Directional Irregular Waves with Side-Wall Reflections. J of the Soc of Naval Archit of Japan 188 251-256 (in Japanese) Umeda N, Hamamoto M, Takaishi Y et al (1995) Model Experiments of Ship Capsize in Astern Waves. J of the Soc of Naval Archit of Japan 177 207-217

Model Experiment on Parametric Rolling of a Post-Panamax Containership in Head Waves Harukuni Taguchi, Shigesuke Ishida, Hiroshi Sawada and Makiko Minami National Maritime Research Institute (NMRI), 6-38-1 Shinkawa, Mitaka, Tokyo, 181-0004, Japan

Abstract Parametric rolling resonance in head waves was investigated experimentally. An experiment using a scale model of a post-Panamax containership was carried out in regular head waves at the 80 metres square basin of NMRI. In the experiment the wavelength, the wave height, the model speed and the encounter angle were widely varied to clarify overall property of parametric rolling resonance in head waves. As a result, conditions in which the head wave parametric rolling resonance is likely to occur and effects of the encounter period, the wave height and the encounter angle on parametric rolling amplitude were clarified. Moreover numerical simulations of roll motion of a ship in pure head waves were also carried out to investigate the property of parametric rolling resonance further.

1 Introduction A post-Panamax containership accident in the north Pacific Ocean, in which that ship suffered extensive loss and damage to onboard containers due to severe parametric rolling in head wave condition (France et al. 2003), led the International Maritime Organization (IMO) to start work to revise the Intact Stability Code (IS Code) in this aspect. Regarding parametric rolling resonance, phenomenon in following waves is well known, while that in head waves has been less studied so far (e.g. Dallinga et al. 1998, Neves and Valerio 2000, France et al. 2003, Neves et al. 2003, Bulian et al. 2003). Therefore the property of behaviour of ships in parametric rolling resonance in head waves has not been clarified thoroughly. In order to establish appropriate safety measures against severe parametric rolling in head wave condition, influence of hull forms, operating condition (ship speed and course in waves), and wave condition (wave length and wave height) on the occurrence of parametric rolling resonance and the resultant magnitude of roll motion should be clarified. M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_15, © Springer Science+Business Media B.V. 2011

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278 H. Taguchi et al.

In this context an experiment with a scale model of a post-Panamax containership was carried out in regular head waves at the 80 metres square basin of NMRI. As a result, some property of the head wave parametric rolling resonance, e.g. conditions in which the phenomenon is likely to occur, have been clarified. In order to examine the property of parametric rolling resonance, some numerical investigation was also carried out.

2 Outline of Model Experiment The experiment was carried out with a free running model in regular waves of various length and height. Using an autopilot device and a motor controller, the encounter angle and the propeller revolution were kept constant for each run.

F.P.

A.P １

9 8

2

7

3

6

4

5

Fig. 2.1 Body plan of model ship

Table 2.1 Principal particulars

Lpp (m) B (m) D (m) d (m) 3 V (m ) Cb GM (m) T (sec.)

Ship 283.8 42.8 24.4 14.0 106,970 0.629 1.08 30.26

Model 3.700 0.558 0.318 0.183 0.237 0.629 0.014 3.460

Model Experiment on Parametric Rolling of a Post-Panamax Containership 279

1.4 1.2 1.0

GZ (m)

0.8 0.6 0.4 0.2 0.0 0.2 0

10

20

30

40

50

60

70

80

90

φ (deg.) Fig. 2.2 Stability curve (GM = 1.08 m)

The model ship is a 1/76.7 scale model of a 6600TEU containership. The main hull up to the upper deck and the forecastle were reproduced in the model, but the deckhouse was ignored. Fig. 2.1 shows the body plan of the model and its principal particulars are shown in Table 2.1. Fig. 2.2 shows the stability curve in the test condition, which was calculated with only the main hull and the forecastle taken into account. In the experiment six degrees of freedom motion, relative water heights at stem, S.S.5 (weather side) and A.E., lateral acceleration beneath the upper deck at S.S.8, rudder angle, number of propeller revolution and speed and trajectory of the model were measured. Table 2.2 Waves used in the experiment / L 0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

5cm

8cm

　 　 ○

○

　 　 　

○ 　 　

Hw 11cm ○ ○ ○ ○ ○ ○ ○ ○ ○

15cm

20cm

○ ○ 　

○

In order to investigate the overall property of parametric rolling resonance, the wavelength , the wave height Hw, the model speed Vm, and the encounter angle  were varied. Table 2.2 summarizes the length and height of waves used in the experiment. The wave height of 11 cm in the model scale corresponds to that of 8.4 m in the actual scale. The encounter angle was varied at every 15 degrees from 135 degrees (bow wave) to 180 degrees (pure head wave). The model speed was

280 H. Taguchi et al.

varied in a wide range so that the critical condition for parametric rolling resonance could be clarified.

3 Experimental Results 3.1 Examples of Parametric Rolling Typical time series of the ship motion and the lateral acceleration at bow in parametric rolling resonance are shown in Figs. 3.1 and 3.2.

φ (deg.)

20 10 0 10 20 0

10

20

30 sec.

40

50

60

40

50

60

50

60

θ (deg.)

4 2 0 2 4 0

10

20

30 sec.

0.4 η (g)

0.2 0.0 0.2 0.4 0

10

20

30 sec.

40

Fig. 3.1 Measured roll angle (top), pitch angle (middle) and lateral acceleration at bow (bottom). /L = 1.6, Hw = 11 cm,  = 180°, and Vm = 0.48 m/s (Fn = 0.08)

Fig. 3.1 shows the measured roll angle  (top), pitch angle  (middle) and lateral acceleration at bow  (bottom) under the conditions of the wavelength to the ship length ratio /L = 1.6, the wave height Hw = 11 cm, the encounter angle = 180 degrees, and the model speed Vm = 0.48 m/s (the corresponding Froude number Fn = 0.08). The ratio of the measured encounter period Te to the natural rolling period T, Te/T is about 0.48. Parametric rolling resonance, where there are two

Model Experiment on Parametric Rolling of a Post-Panamax Containership 281

φ (deg.)

pitch cycles for each one roll cycle, occurs from the beginning. The roll motion immediately becomes in steady state where the amplitude of parametric rolling reaches 19 degrees. And the steady state amplitude of the lateral acceleration beneath the upper deck at S.S.8 is about 0.29 g (including the gravity component).

10 5 0 5 10 15 0

20

40 sec.

60

80

60

80

60

80

θ (deg.)

4 2 0 2 4 20

0

40 sec.

0.4 η (g)

0.2 0.0 0.2 0.4 0

20

40 sec.

Fig. 3.2 Measured roll angle (top), pitch angle (middle) and lateral acceleration at bow (bottom). /L = 1.6, Hw = 8 cm,  = 135°, and Vm = 0.63 m/s (Fn = 0.11)

Time histories of the measured data under the conditions of /L = 1.6, Hw = 8 cm,  = 135 degrees, and Vm = 0.63 m/s (Fn = 0.11) are shown in Fig. 3.2. In this case Te/T is about 0.49. At the beginning the model exhibits a large and small roll responses to every two encounter waves and gradually transits to parametric rolling response. But even at the end of the run, the rolling amplitude seems not to reach steady state.

282 H. Taguchi et al.

3.2 Occurrence of Parametric Rolling Occurrence of parametric rolling resonance in waves of Hw = 11 cm are summarised in Figs. 3.3 and 3.4. χ = 180 deg. 0.65

0.60

Te/ T φ

0.55

0.50 0.45 0.40

0.35 0.00

0.05

0.10

0.15

0.20

0.25

Fn

λ /L = 0.9

χ = 150 deg.

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

0.65

0.60

Te/T φ

0.55

0.50

0.45

0.40

0.35 0.00

0.05

0.15

0.10

0.20

0.25

Fn

Fig. 3.3 Occurrence of parametric rolling resonance. Hw = 11 cm and  = 180° (upper),  = 150° (lower)

Fig. 3.3 shows the occurrence of parametric rolling resonance with the wavelength to the ship length ratio /L as parameter, where the encounter angle  = 180 degrees for the upper figure and  = 150 degrees for the lower one. The horizontal axis is the model speed in Froude number and the vertical axis is the ratio between the encounter period and the natural rolling period in each figure.

Model Experiment on Parametric Rolling of a Post-Panamax Containership 283

The experimental results are indicated on the lines that show the relation between the model ship speed and the encounter period. The symbol of black circle means condition where the steady state parametric rolling resonance (see Fig. 3.1) was observed, the cross means one where parametric rolling resonance was not observed, and the white triangle means one where parametric rolling resonance was observed but did not reach the steady state (see Fig. 3.2). λ/L = 1.2 0.25

0.20

Fn

0.15

0.10

0.05

0.00 135

150

165

180

χ (deg.) λ/L = 1.6 0.25

0.20

Fn

0.15

0.10

0.05

0.00 135

150

165

180

χ (deg.)

Fig. 3.4 Occurrence of parametric rolling resonance. Hw = 11 cm and /L = 1.2 (upper), /L = 1.6 (lower)

Fig. 3.4 summarizes the occurrence of parametric rolling resonance at various encounter angels ( = 135°, 150°, 165°, and 180°), where /L = 1.2 in the upper figure and /L = 1.6 in the lower one. In each figure the horizontal axis is the encounter angle and the vertical axis is the model speed in Froude number. Symbols mean as the same as in Fig. 3.3.

284 H. Taguchi et al.

From these figures we see the followings. (1) Parametric rolling resonance occurs in relatively wide range of the encounter period to the natural rolling period ratio, Te/T, namely Te/T = 0.4 ~ 0.6. (2) In the same waves parametric rolling resonance is more likely to occur in pure head wave condition ( = 180 degrees) than in bow wave condition (= 135 ~ 165 degrees).

3.3 Amplitude of Parametric Rolling 3.3.1 Influence of Encounter Period Fig. 3.5 shows the steady state amplitude of parametric rolling for various encounter periods under the condition of the wave height Hw = 11cm and the encounter angle  = 180 degrees. The horizontal axis is the encounter period to the natural rolling period ratio and the vertical axis is the parametric rolling amplitude s normalised by the wave slope, s/kHw (k is the wave number). 7.0

λ/L =1.0 =1.2 =1.4 =1.6 =1.8 =2.0 =2.2

6.0

φs/kHw

5.0 4.0 3.0 2.0 1.0 0.0 0.30

0.35

0.40

0.45 0.50 Te / Tφ

0.55

0.60

0.65

Fig. 3.5 Influence of encounter period on parametric rolling amplitude. Hw = 11 cm and  = 180°

From this figure we see the followings. (1) The amplitude of parametric rolling changes largely with the variation of the encounter period. (2) For waves of the same length the parametric rolling amplitude becomes the maximum at about Te/T = 0.43 ~ 0.48. (3) The normalised rolling amplitudes for the same encounter period are almost the same except for waves of /L = 2.0 and 2.2.

Model Experiment on Parametric Rolling of a Post-Panamax Containership 285

3.3.2 Influence of Wave Height and Encounter Angle As indicated in Table 2.2, for waves of /L = 1.0, 1.2, and 1.6 measurements with different wave heights were carried out. Fig. 3.6 shows the steady state parametric rolling amplitudes measured under the conditions of  = 180 degrees and Te/T = 0.40 for /L = 1.0, Te/T = 0.45 for /L = 1.2 and 1.6. The horizontal axis is the wave steepness Hw/, and the vertical axis is the normalised rolling angle. In the experiment the rolling amplitude did not become larger as the wave height increased beyond 11 cm. Therefore as indicated in Fig. 3.6 the normalised rolling amplitude becomes smaller as the wave height increases. 8.0 λ/L=1.0 =1.2 =1.6

7.0

φs/kHw

6.0 5.0 4.0 3.0 2.0 1.0 0.0 0.00

0.01

0.02

0.03

0.04

0.05

Hw/λ

Fig. 3.6 Influence of wave height on parametric rolling amplitude.  = 180° and Te/T = 0.40 for /L = 1.0, Te/T = 0.45 for /L = 1.2 and 1.6

Fig. 3.7 shows the influence of the encounter angle on the parametric rolling amplitudes under the conditions of Te/T = 0.45 and /L = 1.2. The normalised rolling amplitudes for bow wave condition ( =150 and 165 degrees) also become smaller as the wave height increases. And the rolling amplitudes for bow wave conditions are smaller than that for pure head wave condition ( = 180 degrees). Investigation into the influence of the wave height on the parametric rolling amplitude is important for judging whether a parametric rolling resonance leads to capsizing in severe waves. In order to draw conclusion of this issue it seems that more data in various conditions (encounter period, wavelength wave height) is necessary.

286 H. Taguchi et al. 7.0

χ =180 (deg.) =165 (deg..) =150 (deg..)

6.0

φs/kHw

5.0 4.0 3.0 2.0 1.0 0.0 0.00

0.01

0.02

0.03 Hw/λ

0.04

0.05

Fig. 3.7 Influence of encounter angle on parametric rolling amplitude. /L = 1.2 and Te/T = 0.45

4 Numerical Investigation In order to investigate the property of parametric rolling resonance further, we carried out numerical simulations with an equation of roll motion of a ship with stability variation in waves. In this paper, as the first step of the numerical investigation, we examine the parametric rolling resonance in pure head waves, where no roll exciting moment acts, and investigate effects of ship speed and wave height on the occurrence of parametric rolling resonance and the resultant rolling amplitude.

4.1 Equation of Roll Motion Roll motion of a ship with stability variation in pure head wave can be generally expressed with equation (4.1)

 I xx  J xx    A  B    W  GZ  ; t   0

(4.1)

where  is the roll angle, Ixx is the moment of inertia, Jxx is the added moment of inertia, A and B are the linear and the quadratic damping coefficients, W is the displacement, GZ(; t) is the time dependent restoring arm in waves. The dots over the symbol represent differentiation with respect to time t. By dividing all terms in equation (4.1) by (Ixx + Jxx), this equation is rewritten to following form

Model Experiment on Parametric Rolling of a Post-Panamax Containership 287

GZ  ; t    2       2 0 GM

(4.2)

where GM is the metacentric height in still water,  = A/2(Ixx + Jxx),  = B/(Ixx + Jxx), and  = WGM/(Ixx + Jxx). The stability variation in waves arises from the relative position of ship to wave G(t), and the vertical ship motions, heave (t) and pitch (t) in waves. Therefore the time dependent restoring arm in waves can be expressed as equation (4.3).

GZ  ; t   GZ  ; G (t ),  (t ),  (t )

(4.3)

GZ is calculated by integrating the hydrostatic pressure over the instantaneous submerged hull under the undisturbed wave profile, which is determined by the relative position of ship to wave and heave and pitch motions. With this calculation method the instantaneous variation of displacement due to the vertical motions is included. In this paper, the vertical motions are assumed to be linear to the wave height and computed with the strip theory. In equation (4.2), the linear and the quadratic damping coefficients obtained from roll decay tests with forward velocity were used. Moreover we examine only steady state response of equation (4.2). Therefore numerical investigation was carried out with calculation method for obtaining bifurcation diagrams (Taguchi et al. 2003).

4.2 Occurrence of Parametric Rolling Occurrence of parametric rolling resonance is directly detected with bifurcation diagram. 4.2.1 Influence of Encounter Period To draw bifurcation diagrams, numerical simulations were conducted for /L = 1.2, 1.6, and 2.0 with the ship speed gradually changed from Fn = 0 to 0.25 (service speed) but the wave height kept constant as Hw = 11 cm. Fig. 4.1 shows the obtained bifurcation diagrams expressed as the function of the resultant encounter period to the natural rolling period ratio, Te/T. In the diagrams the normalised steady state roll responses at Poincaré section, (nT)/kHw, are plotted as much as 20 cycles for every Te/T calculated. Two points at one Te/T means that parametric rolling resonance occurs at such encounter period. In each diagram experimental results are also shown with symbols of the same meaning as in Fig. 3.3.

288 H. Taguchi et al. λ/L=1.2

φ(nT)/kHw

4.0 2.0 0.0 −2.0 −4.0 0.30

0.40

0.50

Te / Tφ

0.60

λ/L=1.6

φ(nT)/kHw

4.0 2.0 0.0 −2.0 −4.0 0.30

0.40

0.50

Te / Tφ

0.60

λ/L=2.0

φ(nT)/kHw

4.0 2.0 0.0 −2.0 −4.0 0.30

0.40

0.50

0.60 Te / Tφ

Fig. 4.1 Bifurcation diagram with changing Te/T. Hw = 11 cm,  = 180°, and /L = 1.2 (top); /L = 1.6 (middle); /L = 2.0 (bottom)

From the top diagram, it is found that for /L = 1.2 the numerical simulation predicts that parametric rolling resonance occurs in wide range of the encounter period to the natural rolling period ratio, namely Te/T = 0.32 (Fn= 0.23) ~ 0.49 (Fn = 0). However as indicated in the diagram, in the model experiment parametric rolling resonance was not observed at Te/T = 0.38. So the numerical simulation with equation (4.2) overestimates occurrence of parametric rolling resonance in short encounter period range. For /L = 1.6 (the middle diagram) it is found that parametric rolling resonance occurs in all the calculated range of parameter. In this case good agreement between the numerical simulation and the experimental results is found. For /L = 2.0 (the bottom diagram) the parametric rolling resonance is predicted to occur in the range of Te/T = 0.45 ~ 0.57. Compared with the experimental results, it is found that the numerical simulation well predicts the lower limit of the encounter periods for the occurrence of parametric rolling resonance, while it underestimates the upper limit.The discrepancies between the numerical simulation and the experimental results in the range of encounter period

Model Experiment on Parametric Rolling of a Post-Panamax Containership 289

which leads to parametric rolling resonance for /L = 1.2 and 2.0 seem to be mainly caused by the prediction of the stability variation in waves. 4.2.2 Influence of Wave Height λ/L = 1.2

6.0

φ(nT)/kHw

4.0 2.0 0.0 2.0 4.0 6.0

0

0.01

0.02

0.03

0.04

0.05

0.06

0.04

0.05

0.06

0.04

0.05

0.06

Hw /λ λ/L = 1.6

φ(nT)/kHw

6.0 4.0 2.0 0.0 2.0 4.0 6.0

0

0.01

0.03

0.02

Hw /λ λ/L = 2.0

φ(nT)/kHw

6.0 4.0 2.0 0.0 2.0 4.0 6.0

0

0.01

0.02

0.03

Hw /λ

Fig. 4.2 Bifurcation diagram with changing Hw/. Te/T = 0.45,  = 180°, and /L = 1.2 (top); /L = 1.6 (middle); /L = 2.0 (bottom)

Fig. 4.2 shows bifurcation diagrams for /L = 1.2, 1.6, and 2.0 with the wave height gradually changed but the ship speed kept constant as the resultant encounter period satisfies that Te/T = 0.45. In each diagrams the horizontal axis is the wave steepness, while the vertical axis is the roll responses at Poincaré section normalised by the wave slope. The experimental results are also shown with the same manner as in Fig. 4.1. From Fig. 4.2 it is found that the numerical simulations are almost consistent with the experimental results for all the wavelengths examined. And it is also found by the numerical simulations that there are two threshold wave heights, the lower limit and the higher limit of wave heights, for occurrence of parametric

290 H. Taguchi et al.

rolling resonance. For /L = 1.2 (the upper diagram), the numerical simulation predicts that as the wave height increases from 0, the parametric rolling resonance appears at about Hw/ = 0.006 and continue to exist up to about Hw/ = 0.045. As the wave height increases further, the parametric rolling resonance disappears. As shown in Fig. 3.6, for /L = 1.2 the model experiment was carried out in waves of up to 20 cm in height (Hw/ = 0.045), and parametric rolling resonance was observed in all the wave height tested. In order to confirm the existence of the higher threshold of wave height for parametric rolling resonance, additional model experiment in higher waves seems necessary. Fig. 4.2 also shows that as the wavelength increases, the range of wave height, which leads to parametric rolling resonance, becomes smaller. As the occurrence of parametric rolling is directly related to the stability variation in waves, further investigation including quantitative examination of the stability variation seems necessary to confirm the effects of the wave height on the occurrence of parametric rolling resonance.

4.3 Amplitude of Parametric Rolling In this section, steady state parametric rolling amplitudes obtained in the calculation of bifurcation diagrams are examined. 4.3.1 Influence of Encounter Period Fig. 4.3 shows the parametric rolling amplitude predicted by the numerical simulations with changing the ship speed as parameter. This figure corresponds to Fig. 4.1. The horizontal axis is the encounter period to the natural rolling period ratio and the vertical axis is the steady state rolling amplitude normalised by the wave slope. In the diagrams the experimental results are shown with the white circles. From Fig. 4.3 the numerical simulation seems to capture the tendency of parametric rolling amplitude found in the model experiment. But except for /L = 2.0, the numerical simulation underestimates the maximum amplitude of parametric rolling and there is also some discrepancy in the encounter period leading to the maximum response. As the stability curve of this ship is hard spring type (see Fig. 2.2), the natural rolling period for large rolling amplitude tends to become shorter than that for small amplitude rolling. The numerical simulation may overestimate this effect and the resultant peak encounter period to the natural rolling period ratio becomes smaller than that in the model experiment.

Model Experiment on Parametric Rolling of a Post-Panamax Containership 291

φs/kHw

φs/kHw

φs/kHw

λ /L=1.2 6.0 5.0 4.0 3.0 2.0 1.0 0.0 0.30

6.0 5.0 4.0 3.0 2.0 1.0 0.0 0.30

6.0 5.0 4.0 3.0 2.0 1.0 0.0 0.30

0.50

0.40

Te / Tφ

0.60

λ /L=1.6

0.50

0.40

Te / Tφ

0.60

λ /L=2.0

0.40

0.60

0.50

Te / Tφ

Fig. 4.3 Influence of encounter period on parametric rolling amplitude. Hw = 11 cm,  = 180°, and /L = 1.2 (top); /L = 1.6 (middle); /L = 2.0 (bottom).

4.3.2 Influence of Wave Height Fig. 4.4 shows the predicted parametric rolling amplitude with changing the wave height as parameter. Compared to the experimental results, which are shown with white circles in the diagrams, the numerical simulation is found to predict the parametric rolling amplitude well. It is also found that the numerical simulation reproduces the tendency shown in the model experiment that the normalised parametric rolling amplitude becomes smaller as the wave height gets larger. In the numerical investigation, as shown in Fig. 4.5, this tendency was observed at different encounter periods too.

292 H. Taguchi et al. λ

1.2

φs

k w

12.0 10.0 8.0 6.0 4.0 2.0 0.0

0

0.01

0.02

0.03

λ

0.05

0.06

0.04 w λ

0.05

0.06

0.04

0.05

0.06

1.6

φs

k w

12.0 10.0 8.0 6.0 4.0 2.0 0.0

0.04

w λ

0

0.01

0.02

0.03

λ

2.0

φs

k w

12.0 10.0 8.0 6.0 4.0 2.0 0.0

0

0.01

0.02

0.03

w λ

Fig. 4.4 Influence of wave height on parametric rolling amplitude. Te/T = 0.45,  = 180°, and /L = 1.2 (top); /L = 1.6 (middle); /L = 2.0 (bottom)

e

φ 0.40

φs

k w

12.0 10.0 8.0 6.0 4.0 2.0 0.0 0.00

0.01

0.02

0.03

e

0.05

0.06

φ 0.50

φs

k w

12.0 10.0 8.0 6.0 4.0 2.0 0.0

0.04 w λ

0

0.01

0.02

0.03

w λ

0.04

0.05

0.06

Fig. 4.5 Influence of wave height on parametric rolling amplitude at Te/T = 0.40 (upper) and 0.50 (lower).  = 180° and /L = 1.6

Model Experiment on Parametric Rolling of a Post-Panamax Containership 293

5 Conclusions The behaviour of a ship in parametric roll resonance in head waves was investigated experimentally with the free running model of a post-Panamax containership in regular waves. And some numerical investigation using an equation of roll motion of a ship with stability variation in waves was also carried out to examine the property of parametric rolling resonance. As a result the followings are clarified. (1) The parametric rolling resonance occurs in relatively wide range of the encounter period, where the ratio of it to the natural roll period, Te/T ranges from about 0.4 to 0.6. (2) In the same wave parametric rolling resonance is more likely to occur in head wave condition ( = 180 degrees) than in bow wave condition ( = 135 ~ 165 degrees). (3) The amplitude of parametric rolling changes largely with the variation of the encounter period and it becomes the maximum at about Te/T = 0.43 ~ 0.48. (4) The parametric rolling amplitude normalised by the wave slope becomes smaller as the wave height increases. (5) The parametric rolling amplitudes for bow wave conditions become smaller than that for the head wave condition. (6) The numerical simulation using equation (4.2) can predict the property of parametric rolling resonance qualitatively. (7) Discrepancies found in both the occurrence range and the resultant amplitude of parametric rolling implies that some refinement for the stability variation model in waves, equation (4.3), is necessary to get more precise prediction.

Acknowledgements Some parts of this investigation were carried out as a research activity of the RRSP4 research panel of the Shipbuilding Research Association of Japan in the fiscal year of 2004, funded by the Nippon Foundation. The authors express their sincere gratitude to the both organizations.

References Bulian G, Francescutto A and Lugni C (2003) On the Nonlinear Modeling of Parametric Rolling in regular and Irregular Waves. Proc of the 8th Int Conf on the Stab of Ships and Ocean Veh 305-323.

294 H. Taguchi et al. Dallinga RP, Blok JJ and Luth HR (1998) Excessive Rolling of Cruise Ships in Head and Following Waves. Proc of RINA Int Conf on Ship Motions & Manoeuvrability, London 1-16. France WN, Levadou M and Treakle T (2003) An Investigation of Head-Sea Parametric Rolling and Its Influence on Container Lashing Systems. Mar Technol 40 1: 1-19. Neves MAS and Valerio L (2000) Parametric Resonance in Waves of Arbitrary Heading. Proc of the 7th Int Conf on the Stab of Ships and Ocean Veh 680-687. Neves MAS, Perez N and Lorca O (2003) Analysis of Roll Motion and Stability of a Fishing Vessel in Head Seas. Ocean Eng 30: 921-935. Taguchi H, Sawada H and Tanizawa K (2003) A Study on Complicated Roll Motion of a Ship Equipped with an Anti-Rolling Tank. Proc of the 8th Int Conf on the Stab of Ships and Ocean Veh 607-616.

Numerical Procedures and Practical Experience of Assessment of Parametric Roll of Container Carriers Vadim Belenky, Han-Chang Yu and Kenneth Weems

Abstract The paper examines several aspects related to practical assessment of parametric roll of container carriers. The numerical procedure is based on application of the Large Amplitude Motion Program (LAMP), the nonlinear potential flow code. Viscous roll damping is included from the roll decay test. The rational choice of the loading conditions is considered. Practical non-ergodicity of the response is taken into account.

1 Introduction The problem of large amplitude roll motions caused by parametric resonance, though known for quite some time, recently made its reappearance in relation to an accident with a large container carrier, (France et al 2003). This paper describes the development of on-board information required by the Guide on Parametric Roll (ABS, 2004). Below is a brief literature review on the related subject. The physical phenomenon of parametric roll was one of the focuses of the 8th International Ship Stability Workshop that was hosted by Istanbul Technical University on October 6th and 7th, 2005 in Istanbul, Turkey. The paper by Neves and Rodríguez (2005) looking into a new mathematical model with nonlinearities defined up to the third order in terms of the heave, roll and pitch couplings is introduced in order to simulate strong roll parametric amplifications in head seas. The influence of hull stern shape is discussed. A theoretical analysis discloses some essential dynamical characteristics associated with the proposed coupled third order mathematical model. The paper by Umeda et al (2005) reviews the latest developments at Osaka University and, among others, includes theoretical prediction techniques for magnitude of parametric rolling. Poincaré mapping and averaging method were then used to obtain the magnitude of parametric roll.

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_16, © Springer Science+Business Media B.V. 2011

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A paper by Ikeda et al (2005) describes a model test carried out to evaluate the effect of bilge keels on parametric roll in beam seas. In the considered case, the amplitude of parametric roll decreased from 25 degrees to almost zero. The paper SLF 48/4/12 considered parametric roll in relation to the new Intact Stability Code. Several important points were made: (1) regular wave case may be more complicated due to nonlinear behavior, (2) regular wave case appears to be more severe than irregular wave case, (3) for the irregular wave case, the probability distribution is not Gaussian. Other relevant papers at SLF-48 were focused on updating of the MSC Circular 707. The paper SLF 48/4/4 contains guidelines for all types of ships, while definition of parametric roll conditions are based on TR≈2TE (TR natural period of roll, TE encounter period of waves) while wave length is between 0.8L and 1.2LThe paper SLF 48/4/8 also suggests that the MSC Circ. 707 guidelines be made applicable to all types of ships as far as parametric roll is concerned. The paper addresses both wave pass effect and pitch coupling and recommends changing speed and heading in order to avoid the encounter period being twice the natural roll period. The papers SLF 48/4/16 and SLF 48/4/17 describe diagrams which are meant to be present on board a vessel. Parametric roll criterion is based on the frequency range 1.9 TE < TR <2.1 TR. Spyrou (2004) examines an approximation of transient solution of a Mathieu equation to estimate how quickly parametric roll rises. The influence of restoring nonlinearity is studied and parametric roll response is found outside of the asymptotic instability zone. The probabilistic aspect includes an envelope presentation, which leads to a formula for joint probability of height of two consecutive waves above the given threshold. The Markov chain could be used to obtain the result for the entire wave group. The paper by Bulian et al (2004) presents the preliminary results of an analytical, numerical and experimental study regarding the problem of non-ergodicity of parametric roll in longitudinal irregular long-crested waves. A series of numerical simulations is performed with 1.5-DOF model of parametrically excited roll motion. Qualitative indications given by the numerical simulations were then compared with experimental tests, showing good agreement.

2 Polar Diagrams as a Form of On-board Information Polar diagrams in wave heading/speed coordinates are a convenient way to present any numerical information for on-board use (see e.g., MSC Circ. 707). Polar diagrams for parametric roll were discussed by Shin, et al, (2003, 2004) and were recommended by ABS (2004). A Generic Polar Diagram for Parametric Roll was proposed by SLF 48/4/4. Polar diagrams for parametric roll considered in this paper are a further development of format (ABS, 2004). They represent the maximum roll angle observed during a one-hour simulation limited by lashing (ABS, 1988) and engine

Numerical Procedures and Practical Experience of Assessment 297

criteria (ABS, 2006). The area of the diagram where maximum roll angle did not exceed these criteria remains blank The area where these criteria were exceeded is color-coded in different types of yellow and red (from 22 to 40 degrees ) in accordance with the value of observed angle. See Figure 1.

Fig. 1 Sample polar diagram, Bretsshnieder spectrum, with significant wave height of 12.5 m and zero-crossing period of 11.5 seconds

3 Intended Use of Polar Diagrams These polar diagrams have been developed as a planning tool and are not intended for use when a vessel is already experiencing extreme roll motions. Real-time use of a polar diagram would be difficult. An experienced sailor is capable of evaluating significant wave height, but nearly no one could tell the exact wave heading with the naked eye. Decreased visibility and nighttime makes estimate of wave parameters even more difficult. Most large container carriers nowadays use a weather routing service that usually includes a weather map that is updated every six hours. When a storm is on the route and a master is considering options, the polar diagrams are expected to be most useful. A weather map would include wave direction, so determining wave heading when the vessel encounters a storm will not be difficult. The speed will be also known as speed loss in waves is also known from operation experience. These two figures will determine the position of the point on the polar diagram corresponding most closely to current loading condition and expected significant wave height and zero-crossing periods of waves.

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4 Numerical Tools for Development of Polar Diagrams Parametric roll resonance is a result of changing stability in waves, so a minimum requirement for any numerical simulation tool is a capability to compute hydrostatic forces on instantaneous submerged surface. Such software would be capable of reproducing the phenomenon, but in order to achieve numerical accuracy sufficient for practical purposes, roll damping has to match experimental results (France, et al 2003). With computational capability currently available for mass use, application of potential hydrodynamic methods seems to be a viable, practical solution. Viscous and vortex-induced forces could be added as external components to facilitate matching of roll damping to experimental results. Two numerical tools which meet these requirements for evaluating parametric roll are the Large Amplitude Motions Program (LAMP) and NLOAD3D codes. Both of these codes are time-domain ship motions and loads prediction codes that are built on a 3-D potential flow solution of the wave-body hydrodynamic interaction problem. Both incorporate an approximate body-nonlinear approach (Weems, et al, 2000), which computes the incident wave forcing and hydrostatic restoring pressure on the actual wetted hull surface while solving for the hydrodynamic disturbance potential on the mean wetted surface using the instantaneous ship velocity and a combined body boundary condition. This approach is sophisticated enough to capture the critical hydrodynamic phenomena associated with parametric roll while remaining fast enough to run the large number of simulations required to build the polar diagrams. Of the two, the LAMP code is the more general implementation of the approach and suitable for multi-hull and high-speed displacement ships as well as conventional monohulls. NLOAD3D has been specifically developed for large commercial ships. Key features of NLOAD3D include an automated procedure for calibrating roll damping based on experimental roll decay data and an integrated module for setting up and running dozens of simulations over a range of speeds, headings, and wave realizations for the polar diagram.

5 Environmental and Loading Conditions Parametric roll, as with any other resonance phenomenon, is quite sensitive to excitation frequency range as well as to natural frequency of the dynamical system. Excitation frequency range is defined by a geographical region of ocean, season and other meteorological and oceanographic factors. The most precise way to present such a range is to use measured or hindcasted wave spectrum in the region of interest. Although precise, such a method is quite expensive, as the set of diagrams then becomes route-specific and unusable for other routes.

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The next best way is to use approximated spectra. Most of them have two parameters related to significant wave height and mean zero-crossing period. These figures could be obtained from wave statistics-based scatter diagrams. Here, an averaged scatter diagram from IACS Recommendation 34 was used. In order to develop set route-independent diagrams, simulations have to be performed for a series of significant wave heights and zero-crossing periods. In lieu of any other data, the simulations are performed for three most probable mean zero-crossing periods for each significant wave height. The calculations start from survival condition with 14.5 m of significant wave height and are repeated for smaller significant wave height (2 meters as a step was used in this case) until large roll motions no longer present a practical problem. The natural roll frequency is most sensitive to GM, therefore, the choice of loading conditions has to cover the entire range of operational values of GM. In lieu of sensitivity analysis, using a smaller step for GM, where the most frequent loading conditions are, seems to be a quite practical idea.

6 Roll Damping Calibration Being potential flow time-domain simulation codes, LAMP and NLOAD3D calculate only the wave component of roll damping, so the rest of the roll damping moment (vortex and friction components) has to be added. At the same time, there are no reliable methods of extracting these contributions from the result of the roll decay test. Based on previous experience (France, et al 2003), the best way to get a realistic simulation with respect to roll damping is to calibrate the code to produce the same results as the roll decay test. Non-potential roll damping moment is accepted in quadratic form:

R( )  B1  B2 |  | 

(1)

Here, B1 is the linear damping coefficient and B2 is the quadratic damping coefficient. Following the standard procedure of processing of results of the roll decay test, a straight line is fitted, using points on the roll decrement chart

D  k  Amp  f

(2)

Here, D is a relative decrement, Amp is roll amplitude, k is a slope and f is an intercept of the fitted line. The calibration procedure for roll damping is essentially a numerical “roll decay test” being repeated for different values of damping coefficients B1 and B2 with the following standard roll decay processing until numerical slope and intercept would match those from the model test. This procedure could be expressed in the form of

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a system of two nonlinear algebraic equations and solved with any appropriate numerical method

k , f   Damp( B1 ( f ), B2 (k ))  0

(3)

Here, Damp is a symbolic expression for a run of numerical simulation followed by a standard procedure of roll decay processing. Symbols B1(f) and B2(k) express that the roll linear damping coefficient depends on intercept while the quadratic roll damping coefficient depends on the slope. Practical experience has shown that the calculations converge after 5-6 iterations, on average.

7 Consideration of Practical Non-ergodicity and Length of Record Practical non-ergodicity of nonlinear rolling has been addressed by a number of authors since the late 1990s (Belenky et al, 1997, 1998, 2001, 2003, Shin, et al, 2004, Bulian, et al 2004) and was found to be especially strong in parametric roll (Belenky 2004). The reason is that waves capable of exciting parametric roll may appear in a different sequence, one after another in one wave record or separated by “harmless” waves in another one. In the last case, kinetic energy may be dispersed before the next portion of parametric excitation will come and, as a result, the roll variance may significantly differ between records. The practical implication of the absence of ergodicity means that several statistically independent wave records have to be used for numerical simulations. Statistical independence of wave records is achieved by using different initial phases. Another aspect to be addressed is the length of a single record. As is well known (see, for example, Belenky and Sevastianov, 2007), the length of a wave record modeled by the inverse Fourier Transform is related to the frequency step. If this length is exceeded, the record can no longer be considered as statistically representative. The reason for the effect is an error in the numerical integration (Belenky 2005, se also the Chapter “On Self-Repeating Effect in Reconstruction of Irregular Waves” in this book). Evidently, this length could be increased by decreasing the frequency step. However, it will lead to an increasing number of wave components necessary for correct presentation of the wave spectrum. Increasing the number of wave components may lead to a significant increase in the cost of the simulation as more components have to be added up at every time step. Based on the observation, an attempt to increase the length of the simulation twice leads to a four times increase in calculation time. At the same time, several relatively short records do have an advantage in comparison with a fewer long ones. They not only save computation time but take care of the practical non-ergodicity problem. Obviously, statistical methods should

Numerical Procedures and Practical Experience of Assessment 301

be used to estimate the accuracy that could be achieved by different combinations of number and length of records.

8 Some Aspects of Topology of the Polar Diagram As the polar diagram (such as shown in Fig. 1) is a result of numerical simulation, all large amplitude roll motions are included (coming from both parametric and synchronous resonance). However, a simple analysis of frequencies could reveal motions apparently coming from synchronous resonance. One polar diagram is generated for a specific sea state defined with significant wave height and mean zero-crossing period. Each of the two-dimensional grid points of the polar diagram represent wave heading and ship speed. The encounter wave period can be determined using the mean zero-crossing period, wave heading and ship speed for each grid point. Large roll angles are often observed around the grid points where the encounter wave period is close to the roll natural period. This is typically due to the roll synchronous resonance rather than parametric roll response, which is mostly to occur near the grid points where the encounter period is one half of the roll natural period. For example, the large roll angle peaks around 20-25 knots speed near 60° and 300° headings in Fig, 1 are typically due to roll synchronous resonance. The most evident feature related to parametric roll is that, generally, the maximum angle is decreased with the increase of speed. This could be explained by the fact that roll damping increased with the speed raises the threshold of sensitivity to parametric excitation, so a lesser number of waves can transfer energy. It is noticeable that for many such diagrams, the maximum of the parametric roll does not necessarily fall exactly in head seas. At the same time, head and following seas are expected to cause the most difference in stability between crest and trough of the wave. Shifting of maxima is also less noticeable for smaller speeds. Such a shift of maximum can probably be explained by changes in encounter spectrum. As is well known, in head encounter, the spectrum becomes wider (see Fig. 2a, where four encountered spectra are shown, corresponding to 90, 120, 150, 180 degrees of wave heading). As the encounter spectrum becomes broader, less energy is concentrated in the frequency range where the dynamical system is susceptible to parametric excitation. There are two competing factors; with 180 degrees of heading approaching, the dynamical system becomes more susceptible to parametric excitation due to increased wave influence on stability, and there is less parametric excitation due to the form of the encounter wave spectrum.

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20

b)

a) Encounter Spectrum

200

o

90

Encounter Spectrum 0o

10

150o

o

30o

120

100 180o

60o

, 1/s 0

0.5

1

1.5

0.2

0.3

0.4

90o 0.5

0.6

, 1/s 0.7

Fig. 2 Transformation of encounter spectrum at 15 knots significant wave height 8.5 m

Wave direction has a smaller influence on encounter spectrum for smaller speeds; therefore, there is not as much of a shift of maximum for slower speeds. A similar physical mechanism is responsible for topology in following seas. For the slower speeds and headings close to 90 degrees, the encounter spectrum becomes narrower (Fig. 2b), but the system already can take parametric excitation. As the concentration of energy is high (the encounter spectrum is narrow), severe parametric roll is observed. However, these conditions do not exist in the large area in the sample diagram in Fig. 1 because decreasing heading and increasing speed leads to the appearance of negative encounter frequencies and even more narrow encounter spectrum. Apparently, the range of susceptible frequencies of the dynamical system is different from the frequency range of encounter spectrum, and the parametric roll mode very quickly ceases to exist. For the cases of smaller GM, (see sample diagram in Fig. 3), a significant area with parametric roll is also possible in following seas. Topology of such a diagram is generally similar to the one observed above and could be explained by the same physical reasons. Parametric roll in following seas is more likely because the low GM range of sensitivity to parametric excitation is shifted towards lower frequencies and following seas can match it.

Numerical Procedures and Practical Experience of Assessment 303

Fig. 3 Polar diagram - relatively small GM Bretschnieder spectrum, with significant wave height of 8.5 m and zero-crossing period of 11.5 seconds

9 Summary The polar diagram is based on the maximum roll angle observed during a one-hour simulation. Criteria for polar diagrams are based on lashing strength and engine conditions. Exceeding of these criteria is color-coded, depending on the value of the roll angle. Simulations for polar diagrams are carried out using potential codes calibrated for roll damping with roll decay tests. Polar diagrams have complex topology, primarily influenced by encounter spectrum of waves and natural frequency. The choice of loading conditions should be determined by sufficient coverage of the range of natural frequency.

Acknowledgements The authors wish to express their appreciation and gratitude to the management of the American Bureau of Shipping and Science Application International Corporation. The development of the LAMP System has been supported by the U.S. Navy, the Defence Advanced Research Projects Agency (DARPA), the U.S. Coast Guard, ABS, and SAIC. LAMP development has been supported by the Office of Naval Research (ONR) under Dr. Patrick Purtell. The development of NLOAD3D program has been supported by ABS.

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References American Bureau of Shipping (1988) Guide for Certification of Container Securing Systems, N. Y. American Bureau of Shipping (2004) Guide for the Assessment of Parametric Roll Resonance in the Design of Container Carriers, Houston, Texas American Bureau of Shipping (2006) Rules for Building and Classing Steel Vessels, Houston, Texas Belenky V (2004) On Risk Evaluation at Extreme Seas. Proc of 7th Int Ship Stab Workshop, Shanghai Belenky V (2005) On Long Numerical Simulations at Extreme Seas. Proc of 8th Int Ship Stab Workshop, Istanbul Belenky V, Degtyarev AB, Boukhanovsky AV (1998) Probabilistic Qualities of Nonlinear Stochastic Rolling Ocean Eng 25(1):1-25 Belenky V, Suzuki S, Yamakoshi Y (2001) Preliminary Results of Experimental Validation of Practical Non-Ergodicity of Large Amplitude Rolling Motion, Proc of 5th Int Stab Workshop¸ Trieste Belenky V, Sevastianov NB, (2007) Stability and safety of ships. Risk of capsizing, 2nd ed. SNAME, Jersey City Belenky V, Weems KM, Lin WM, Paulling JR (2003), Probabilistic analysis of roll parametric resonance in head seas, Proc of 8th Int Conf on Stab of Ships and Ocean Veh, Madrid Bulian G, Lugni C, Francescutto A (2004) A contribution on the problem of practical ergodicity of parametric roll in longitudinal long crested irregular sea Proc of 7th Int Ship Stab Workshop, Shanghai France WM, Levadou M, Treakle TW, Paulling JR, Michel K, Moore C (2003) An Investigation of Head-Sea Parametric Rolling and its Influence on Container Lashing Systems. Marine Technol¸ 40(1):1-19 IACS Recommendation No 34, Standard Wave Data Ikeda Y, Munif A, Katayama A, Fujiwara T (2005) Large Parametric Rolling of a Large Passenger Ship in Beam Seas and Role of Bilge Keel in Its Restraint Proc of 8th Int Ship Stab Workshop, Istanbul Neves MA, Rodríguez CA (2005) Stability Analysis of Ships Undergoing Strong Roll Amplifications in Head Seas Proc of 8th Int Ship Stab Workshop, Istanbul Shin YS, Belenky VL, Lin WM, Weems KM, Engle AH (2003) Nonlinear Time Domain Simulation Technology for Seakeeping and Wave-Load Analysis for Modern Ship Design SNAME Trans 111 Shin YS, Belenky VL, Paulling JR, Weems KM, Lin WM (2004) Criteria for Parametric Roll of Large Containerships in Longitudinal Seas, SNAME Trans 112 SLF 48/4/4 Review of MSC/Circ.707 to include parametric rolling in head seas (Australia and Spain) SLF 48/4/8 Proposed Revision of MSC/Circ.707 (to include parametric rolling in head seas) (Germany) SLF 48/4/12 On the development of performance-based criteria for ship stability in longitudinal waves (Italy) SLF 48/4/16 Proposal on MSC/Circ.707 revision (Russian Federation) SLF 48/4/17 Proposal on MSC/Circ.707 revision (Russian Federation) Spyrou KJ (2004) Criteria for parametric rolling, Proc of 7th Int Ship Stab Workshop, Shanghai

Numerical Procedures and Practical Experience of Assessment 305 Tikka KK, Paulling JR (1990) Prediction of Critical Wave Conditions for Extreme Vessel Response in Random Seas. Proc of 4th Int Conf on Stab of Ships and Ocean Veh, Naples Umeda N, Hashimoto H, Paroka D, Hori M (2005) Recent Developments of Theoretical Prediction on Capsizes of Intact Ships in Waves. Proc of 8th Int Ship Stab Workshop, Istanbul Weems KM, Lin WM, Zhang S, Treakle T (2000) Time Domain Prediction for Motions and Loads of Ships and Marine Structures in Large Seas Using a Mixed-Singularity Formulation, Proc of the Fourth Osaka Colloquium on Seakeeping Performance of Ships, Osaka

Parametric Roll and Ship Design Marc Levadou and Riaan van’t Veer Maritime Research Institute Netherlands (MARIN)

Abstract The variety of vessels which can suffer from parametric roll is large. It has been observed on small fishing vessels sailing in following waves but also on large cruise and container vessels in head and following waves. The occurrence of parametric roll, with high roll angles, is governed by a complex combination of main dimensions, loading condition, hull form, appendage configuration, speed and encountered wave conditions. In this paper the influence of main dimension and of variations on the fore and aft body on the occurrence of parametric roll are investigated. A single degree of freedom motion method and a nonlinear time domain simulation method were used. The results were validated with model tests on a C11 container ship. Also, the influence of different roll damping devices on the occurrence of parametric roll is evaluated.

Nomenclature Axx

bc B Bxx Btotal Bcrit Cxx

C Dbk C FH C FH C L  g GMT

: roll added mass : damping gain : ship beam : roll damping : total damping : critical damping : restoring : effective drag coefficient : magnification of the flow over the bilge coefficient : fin to the hull coefficient : lift slope : gravity : transverse metacentre height

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_17, © Springer Science+Business Media B.V. 2011

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hBK

: bilge keel height H : wave height Hs : significant wave height Ixx : ship inertia in roll kxx : radius of gyration of roll L or L pp : length between perpendiculars

l BK Mx

rBK rfin

: bilge keel length : first order wave excitation : arm of bilge keel to centre of gravity : arm of fin stabilizer to centre of gravity

SGM S e

: spectral density of the wave encounter

T

: roll natural period

Tp

: wave peak period : encounter wave peak period

T pe Vs

 

ρ



 a  

: spectral density of GM fluctuations

: ship speed : roll motion : roll velocity : density of water : displacement : natural roll frequency wave : amplitude : non dimensional damping reduction : wave length

1 Introduction In 1998 a post-Panamax, C11 class containership lost 1/3 of her deck containers and damaged another 1/3 in a severe storm. The incident was analyzed by means of numerical simulations and model tests. The results confirmed that the vessel suffered from a severe case of parametric roll during the storm (France 2003). Since the publication of these results, ship operators and ship designers have become more aware of the fact that this phenomenon can occur for larger vessels in confused seas and not only for small vessels in regular waves which was thought for many years.

Parametric Roll and Ship Design 309

New designs of large vessels, in particular container vessels, are since then more and more checked on their tendency for parametric roll behaviour. Class societies acknowledged the problem and have started incorporating parametric roll in their guides (Shin 2004). Still, model tests are considered as the way to assess the sensitivity for parametric roll and such tests are expensive in a design parameter space. Analytical tools that can predict the phenomenon are not easy to use, not always reliable or available. Therefore hull lines variations or optimization with regard to parametric roll is seldom performed in an early stage of new designs. In order to incorporate parametric roll in ship design an understanding of the phenomenon is required. A greater understanding is needed in how main dimensions, hull form changes and appendages configuration alter the probability of parametric roll. In this paper result of a study on the effect of main dimension variations, hull form variations and different appendages configuration on the occurrence of parametric roll will be presented. In Chapter 2 the phenomenon is explained and a discussion of the minimal requirements for parametric roll to occur is given. In the third chapter a discussion will be given of how the main dimensions and the loading condition of a vessel can result in conditions where parametric roll can occur. Different graphs are presented showing the “critical” combinations of vessel length and loading condition. In Chapter 4 results of the effect of hull from variations on a C11 post-panamax container vessel on the occurrence of parametric roll are presented. Variation of bow flare and stern configuration is investigated. In the last Chapter the effect of different roll reduction devices such as bilge keels, anti roll tanks and active fin stabilizers is discussed.

2 Background 2.1 Theory The theory behind parametric roll and its consequence has been studied and described by many investigators, see for example Kempf (1938), Graff and Heckscher (1941), Paulling (1959 and 1961), Oakley (1974), Dunwoody (1998), Dallinga (1998), or Luth (1998). Therefore, in this paper only the principles of parametric roll will be described. In “normal” sailing conditions the ship motions of a vessel are caused by direct wave excitation. Resonant roll motion often occur in beam waves and stern quartering waves, when the combination of wave period, vessel speed and heading leads to an encounter frequency wave period close to the natural roll period of the vessel. In pure head seas condition, the first order roll wave excitation is zero. Nevertheless, under certain conditions of encounter period, roll motion can be excited in head seas, via a different phenomenon. This phenomenon is referred to as “auto

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parametrically excited motion” which is usually shortened to “parametric motion” or “parametric roll”. The term describes a state of motion that results not from direct excitation by a time-varying external force or moment but from the periodic variation of certain parameters of the oscillating system. The roll motion, once started, may grow to large amplitude, limited by roll damping and, in extreme conditions, may result in danger to the ship or its contents. For a ship in head or stern seas the uneven wave surface together with the pitch-heave motion of the ship results in a time-varying underwater hull geometry. This varying geometry, in turn, results in time-varying changes in the metacentric height, i.e., in the static roll stability. The variation of the static roll stability can cause instability if it occurs in the appropriate period. From theory and as validated by model tests (Dallinga 1998, Luth 1998, France 2003), parametric roll occurs when the following requirements are satisfied:  The natural period of roll is equal to approximately twice the wave encounter period.  The wavelength is in the order of the ship length (between 0.8 and 1.2 times LBP).  The wave height (thus the GM variations) exceeds a critical value.  The roll damping is low (lower threshold wave height).

2.2 Prediction of Parametric Roll Nonlinear, time domain seakeeping computer codes are able to predict the phenomenon of parametric roll (see below). These computations are however not easy to use or available for everyone. In a preliminary design stage a simple and fast method is desirable. For parametric roll Dunwoody (1989) proposed such a method, which is used in this paper. His method is based on a single degree of freedom motion equation for roll, using a time varying restoring coefficient. This motion equation is known as the Mathieu equation, which is presented first. 2.2.1 Modeling in One Degree of Freedom The equation for one degree of freedom roll motions is given by equation (1): d 2 d (1)  I xx  Axx  2  Bxx  Cxx  M x dt dt In head seas condition the roll moment excitation will be zero, similar as in a roll decay test. The roll motion equation reduce in these situations to: d 2 d  2  2  0 (2) 2 dt dt where the roll period is defined by:

Parametric Roll and Ship Design 311

 

Cxx  I xx  Axx ( )

 g GM I xx  Axx ( )

and where the damping ratio is defined as: Bxx ( )  2  I xx  Axx ( ) 

(3)

(4)

The restoring moment or static stability GM of the ship when sailing in waves will vary in time and the variation is a function of the actual wetted surface contour and thus depends on the hull lines around the calm water line. The largest variations in restoring occurs when the ship has large pitch motions, so when it sails in a wave length equal to about the ship length. A formula for the time varying restoring force, for the upright ship, in regular waves is given by: Cxx (t )   g   GM m  GM a cos( t )  (5) Where GM m is the mean GM, that is the GM in calm water GM a is the amplitude of the GM variation and  is the wave frequency. When equation (5) is used in equation (1) a one-dimensional Mathieu equation is obtained; a linear second order differential equation with periodic coefficients. The damped Mathieu equation is now written as: d 2 d  2  2  a cos( t )    0 (6) 2 dt dt Parametric resonance conditions for this equation can be found when:   1 (7) n  2 2 2 According to Francescutto (2002) the threshold value for parametric roll in regular waves is:  GM 4 (8)  GM n This threshold has been used and compared with simulations and model test. The results are presented in section 4.4. Furthermore in this paper a one degree of freedom based method proposed by Dunwoody (1989) will be used. The method is shortly described below. 2.2.2 Response to GM Fluctuations (Dunwoody, 1989) The method proposed by Dunwoody is based on the following assumptions: 1. The roll motion can be expressed by the differential equation for a single degree of freedom oscillator with parametric excitation of the stiffness. 2. That there is a linear relation between roll stiffness excitation (GM fluctuations) and wave height.

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3. That a relation can be made between the spectrum of the stiffness fluctuations with the incident wave height spectrum and the speed of the ship. 4. That GM fluctuations produce an effect analogous to a reduction in the roll damping. 5. If the reduction in roll damping is larger than the roll damping from the hull and appendages an unstable situation occurs and the vessel is subject to parametric roll. Furthermore, Dunwoody demonstrates that for a wide band random process the spectral density can be estimated for a frequency of twice the natural roll frequency. Given the assumptions above the spectral density of the GM fluctuations can be expressed as the product of the spectral density of the wave encounter spectrum (at twice the natural roll frequency) and the square of the transfer function of the GM fluctuations (at twice the natural roll frequency). This is expressed in equation (9). 2

 GM a ( )  (9) SGM    Se ( ) a   According to Dunwoody the non dimensional damping reduction follows from:  g 2 SGM (10)   43 k xx4 Parametric roll will occur if the non dimensional roll damping reduction exceeds the total roll damping Btotal (made dimensionless by dividing with the critical roll damping Bcrit ), see equation (11).

Btotal    0 Bcrit Bcrit  2

(11)

 Axx  I xx   g GM

The threshold wave height (the wave height for which parametric roll will start) can be determined by varying the wave spectrum in equation (9) and determining when equation (11) becomes zero. 2.2.3 Nonlinear, Time Domain Seakeeping Code PRETTI The development of a 3D panel code for seakeeping motion prediction has been a point of interest for the Co-operative Research Ships (CRS) since many years. In this joint-industry project a large group of different companies, such as class societies, ship-yards, ship operators, navies and research/engineering companies are actively involved in research related to many aspects in the design and operation of ships. In recent years the CRS has developed a time-domain seakeeping code (Pretti) based on the hydrodynamics as calculated in the frequency domain by a 3D panel code (Precal). The current time-domain code incorporates nonlinear excitation by pressure integration over the actual wetted surface. Diffraction forces are considered

Parametric Roll and Ship Design 313

linear. Hydrodynamic coefficients and oscillatory (manoeuvring) derivatives are specific to sinusoidal motions and in a general theoretical model a ship manoeuvring in a seaway, the ship motion cannot be considered simply sinusoidal. The motion equation must account for transient and random motions. This problem was initially discussed by Cummins (1962) and this approach of impulse response functions is adopted in Pretti. The behaviour or a ship travelling in a seaway ‘integrates’ in practice two areas which are traditionally studied as separate problems and have lead to two different mathematical approaches: a seakeeping theory assuming small motion amplitudes and the theory of manoeuvring assuming calm water and thus frequency independent hydrodynamics. In Pretti the two theories are combined as discussed by Bailey (1997) or Fossen (2004). This is a challenging area of research since there is an overlap between the two models, which require a careful implementation of manoeuvring coefficients. Ideally the aim is that with vanishing wave height the manoeuvring capabilities of the ship are found, and that the seakeeping hydrodynamics are captured in moderate wave conditions. In large wave conditions with large amplitude motions the assumptions behind both the seakeeping and manoeuvring theory are violated since large variations in wetted surface are not accounted for when the basic coefficients in the models are calculated. Model tests are an essential guidance for the user of nonlinear time domain simulation tools to gain experience in the use of a unified model. The current nonlinear time domain code of CRS focuses on course-keeping of the ship in 6 degrees of freedom (6DOF), in which the interaction with the manoeuvring model is already essential. Especially the sway hydrodynamics will influence roll motions and this means that seakeeping (roll damping) and manoeuvring is to be combined. Yaw manoeuvring and a PID controlled rudder are further more essential to keep the ship at its track and course.

3 Influence of Main Dimensions on Parametric Roll In the theoretical background four criteria for parametric roll to occur were given. The first two (the natural period of roll is equal to approximately twice the wave encounter period and the wavelength is on the order of the ship length) can be described with the following equations: T  2Tpe (12)

  LPP

(13) Equation (14) gives an approximation for the natural roll period (assuming 10% added mass). Equation (15) gives a relation between wavelength (λ) and wave period for deep water.

314 M. Levadou and R. van’t Veer

T 

2.2 k xx

(14)

gGM 2 g

Tp 

(15)

For zero speed equations (14) and (15) can be substituted in (12). GM can then be expressed in the following way (for zero speed).

GM  0.605

k xx2

(16)



For non zero speed the following relation between peak wave period and peak encounter wave period should be used (Vs being the ship speed and μ the wave direction, 180 being head waves).

Tpe 

Tp 2 Vs cos  1 Tp g

(17)

Equation (16) gives a relation between GM, radius of gyration and wave length. By substituting also the second criteria for parametric roll (equation (13)) the relation becomes purely dependent of geometry and vessel main dimensions. When the radius of gyration is not known, standard values for kxx/B can be used. By also using standard values for L/B equation (16) can be rewritten in the following way:

GM  0.605

L2  k xx    L 

2

(18)

In Table 1 the values for kxx/L are given for a range of L/B and kxx/B values. Table 1 kxx/L for a range of L/B and kxx/B values L/B kxx /B

5.6

5.8

6

6.2

6.4

0.38

0.0679 0.0696 0.0714 0.0732 0.0750

0.0655 0.0672 0.0690 0.0707 0.0724

0.0633 0.0650 0.0667 0.0683 0.0700

0.0613 0.0629 0.0645 0.0661 0.0677

0.0594 0.0609 0.0625 0.0641 0.0656

0.39 0.40 0.41 0.42

Using equation (18) and the above table, two graphs, Fig. 1 for zero speed and Fig. 2 for 10 knots, can be made. They show the combinations of GM and Lpp for which parametric roll instability might occur. In other words where the first two criteria for parametric roll to occur are met.

Parametric Roll and Ship Design 315

Fig. 1 Combinations of GM and vessel length resulting in parametric instability at 0 knots

Fig. 2 Combinations of GM and vessel length resulting in parametric instability at 10 knots

Fig.1 and Fig 2. show the relation between GM and Lpp for an upper limit of wavelength/Lpp = 1.2 (dashed lines) and a low limit of wavelength/Lpp = 0.8 (solid lines). For a certain kxx/Lpp ratio the area between the dashed and solid lines give the GM range for a certain Lpp for which the criteria are met. With these graphs it is easy to check in an early stage if the vessel can be subject to parametric roll. To further illustrate how to use these graphs an example is given below. C11 class container vessel: Lpp = 262 m; B = 40 m; kxx/B = 0.40; Thus: kxx/L = 0.061 In Fig. 1 and Fig. 2 the range of GM’s which will give parametric instability is indicated. At zero speed the range will be between GM = 1.5 m and GM = 2.2 m and at 10 knots speed between GM = 2.2 m and GM = 3.8 m. This doesn’t mean the vessel will actually be subject to parametric roll. This depends on other factors

316 M. Levadou and R. van’t Veer

like the amount of parametric excitation (GM fluctuations in waves) and the amount of roll damping. These factors depend on the hull form and the size of the appendages.

4 Hull Form Variations 4.1 Selected Case In order to investigate the influence of the hull form on the occurrence of parametric roll a hull form needed to be selected. As a case the post-Panamax, C11 containership which encountered the storm, as described in the introduction, is used. It is a logical starting point since model tests were previously performed on the same vessel at MARIN.

Fig. 3 C11 Hull Form

A rendering of the C11 hull form is presented in Fig. 3. Typical of postPANAMAX containerships, the C11 has extensive bow and stern flare. The ship’s natural roll period as used in this study was 25.7 seconds. This corresponds to the estimated roll period of the ship at the time of the incident. The GM in calm water was about 2 m. The original hull form and variations at the bow and aft of the hull form were investigated. In the aft a Pram type hull form and a hull form with higher deadrise were taken. At the bow hull lines with a less pronounced bow flare (55 deg) and a more pronounced bow flare (45 deg) were used. This resulted in four variations of the original hull form. In order to have a fair comparison between the different variations it is necessary to keep the draft, GM and natural roll period the same. Because of the different under water hull form it means that the displacement and KG will vary slightly between the different hull variations.

Parametric Roll and Ship Design 317

4.2 Model Tests In order to have validation material model tests were performed. These were performed in the Seakeeping and Manoeuvering Basin (SMB) at MARIN. The basin measures 170 x 40 x 5 m in length, width and depth respectively. Not all hull form variations were tested. The original hull form (model 8004-1) and the pram aft body (model 8004-2) variation were tested.

Fig. 4 Original C11 (upper) and Pram type (lower) aft ship

318 M. Levadou and R. van’t Veer

The models were self propelled during the tests and completely free sailing. The only connection between the model and the carriage consisted of an umbilical for power and data transmission. The model tests were performed for an average speed in waves of 5 knots. The tests were done for head sea conditions and a variation of wave heights and wave periods were tested. For each model the tests started with a test in high waves (Hs = 7.5 m) for which several wave periods were tested. This was done to determine the most critical wave period. For this critical wave period a series of tests with increasing wave heights were done. For all these tests the realization of the encountered waves was kept the same. Only the height of the waves was increased. The same realization of the waves was used for both the models. This means that wave group effect do not influence the comparison of the two vessels. From the model test results it is possible to determine the wave heights for which parametric roll will start (threshold wave height).

4.3 Model tests Results In Fig. 5 the roll damping of the two hull forms (determined from roll decay tests) is compared. One can see that the roll damping of the pram aft shape is slightly higher, which is according to expectations. Both models were equipped with 40 cm high 76.54 m long bilge keels.

Fig. 5 Roll damping comparison

In Fig. 6 a sample of roll motion time traces are shown for the C11 original hull form in head seas at 5 knots speed. Results for two wave heights are given. They show the roll time trace for the vessel within the same wave realization (same wave

Parametric Roll and Ship Design 319

Fig. 6 Time trace roll motion of C11 Original hull form

but only different amplitude). As can be seen, although the difference in wave height is small (Hs = 4.3 m versus Hs = 3.65 m) the difference in roll response is big. The results illustrate quite well the threshold behaviour of parametric roll. The model tests results are summarized in Fig. 4-5. The figure shows the mean of the 1/10 highest roll motions (A1/10+) as function of the wave height. Each dot in the figure represents a test in irregular head seas.

Fig. 7 Model test results

The model tests results show that the pram hull form results in higher roll angles. It also shows that the roll motions due to parametric roll start earlier meaning that the threshold wave height is lower. From Fig. 5 it could be seen that the damping of the pram is slightly higher than the original v-shape. The results presented in Fig. 7 however show that the roll response is higher. It means that the excitation (GM variations) of the pram hull

320 M. Levadou and R. van’t Veer

form is higher which is also according to expectations. The difference in GM variations will be discussed in the next section. Using a criterion for parametric roll of 10 deg A1/10 one can determine the threshold wave height from Fig. 7. For the C11 original hull form the threshold wave height becomes Hs = 4.3 m and for the pram Hs = 3.5 m.

4.4 Approximation Methods The first step of the calculations was to validate the different methods. If a good enough agreement was found with the model test the second step was to perform calculations for the other hull form variations which were not tested. Calculations were performed using a simplified one degree of freedom method (Dunwoody) and non linear time domain program PRETTI (see also section 2.2). GM variations As discussed earlier in section 2.2 the GM variations in head waves represent the excitation for parametric roll. The static stability in “frozen” longitudinal waves is a good indication of the GM variations of a vessel sailing in waves. These were calculated for the original C11 hull form and the different variations. For this purpose the program SHCP (NSSC 2003) was used. The input for the program are the hull form, loading condition, wave height, wave length and the longitudinal position of the wave crest with respect to the hull. A range of wave conditions and roll angles can be entered. For each condition the pitch-heave static equilibrium is solved (thus preserving equilibrium of weight, buoyancy and trim moments). The righting arms calculated for each condition can be used to determine the GM and thus the GM difference between the sagging and hogging conditions. Using this method one assumes that the dynamic pitch and heave motion do not influence the GM variations. Sample results are given in the Fig. 8. From Fig. 8 the difference in GM between the sagging in hogging conditions can be determined. This can be done for different wave lengths.

Fig. 8 Curve of static stability C11 class container vessel

Parametric Roll and Ship Design 321

In Fig. 9 the GM variations for a range of wave lengths are given for a wave height of H = 5.0 m. The figure shows the GM in sagging condition, hogging condition, the difference between the two (Delta GM) and the difference divided by the wave height (Delta GM/H) which is plotted on the second y-axis (right).

Fig. 9 GM and GM variations versus wave length for C11 class container vessel

In Fig. 10 the linearized GM variations (DeltaGM/H) is given as function of wave frequency for the C11 hull form and the four variations. The frequency corresponding to a wave height equal to the length of the vessel is indicated in the figure.

Fig. 10 Influence of hull shape of GM variations

The following observation can be made from the results presented in Fig. 10. First it can be observe that the changes to the bow flare do not seem to influence the GM variations. Secondly, the V shape aft hull form shows smaller GM variations compared to the original C11 hull form and the pram aft shape shows larger GM variations as the C11 hull form. The latter is in accordance with the model tests results presented in Section 4-3.

322 M. Levadou and R. van’t Veer

Threshold value for parametric roll in regular waves Using equation (8) an estimate of the required GM fluctuations in regular waves can be made. This was done using the following data at 5 knots forward speed: GM = 2.0 m; Bxx = 2.5 e5 Nms/rad, Ixx = 2.12 e7 kNms2; Axx= 3.0 E5 kNms2, which leads to δGM=0.16 m in regular waves of 12.8 s. This is the threshold value at zero roll amplitude. Model tests in regular were not performed with the C11, so that these values cannot be verified. However, compared to the observations in irregular seas, the GM fluctuations are about a factor 10 larger before the C11 shows a steep increase in roll. Response to GM fluctuations (Dunwoody) The results presented in Fig. 10 were used in Dunwoody’s method (see formula 7, 8 and 9). For the damping the results from roll decay tests (see Fig. 5) were used. It was assumed that the different hull form variations had the same roll damping as the original C11 hull form. For each hull form variation the threshold wave height was determined for a speed of 5 knots and a wave period of Tp = 14 s. The results are given in Table 2. Table 2 Comparison threshold wave height Dunwoody and Model tests. Threshold wave height for parametric roll Dunwoody

Model tests

C11 original

4.3 m

4.3 m

Pram aft

3.2 m

3.5 m

V shape aft

5.3 m

-

Large bow flare (45 deg)

4.3 m

-

Small bow flare (55 deg)

4.3 m

-

The results using Dunwoody’s method show very good comparison with the model tests performed. Furthermore the results from Dunwoody’s method confirm the results found from the GM variation calculation. The bow flare does not seem to have any influence on the threshold wave height. The shape of the aft of the vessel has some influence. A very pronounced V shape aft gives the best performance with regard to parametric roll. It must be mentioned that although the bow shape does not directly influence the occurrence of parametric roll it can indirectly influence it. A more pronounced bow flare will results in more slamming events which in turn will make a vessel master decide to reduce the speed earlier than a vessel with a less pronounced bow flare. Because the roll damping at low speed is smaller it is generally more vulnerable to parametric roll than at high speed. So, indirectly the bow flare does have influence o the occurrence of parametric roll. With Dunwoody’s method it is very easy to perform variations in wave conditions. The threshold wave height can then be determined for a variation of wave periods. By combining these lines with a wave scatter diagram it is then possible to evaluate

Parametric Roll and Ship Design 323

the effect of the hull form on the probability of occurrence of parametric roll. This is shown in Fig. 11.

Fig. 11 Threshold wave heights within North Atlantic scatter diagram

In Fig. 11 the limiting wave height is shown for the original C11 hull form and the two aft shape variations (pram and sharp V). The probability of each combination of Hs and Tp is given in thousands. The wave scatter diagram used was area 9 (North Atlantic) all directions from the Global Wave Statistics (Hogben 1986). From Fig. 11 it can be determined that the V shape aft hull form has a 10% smaller number of occurrences of parametric roll (at 5 knots speed in head waves) in the North Atlantic. The results presented in the figure cannot be used to estimate the exact number of occurrences of parametric roll for a given ship. In order to make that estimation the probability of the speed, wave heading and loading condition need to be incorporated. A method to determine this was presented by the author in 2003 (Levadou 2003).

4.5 Simulations Methods The nonlinear time domain simulations have been performed in 5DOF (sway, heave, roll, pitch, yaw) for the lower sea states and in 3 DOF (heave, roll, pitch) for the more extreme sea states. It was verified that the results for intermediate sea states were comparable. The reason to limit the number of DOF in large sea states is a practical one: the Pretti code does not account for large yaw variations. Besides, in larger sea states the numerical model is more difficult to control, in particular due to the relative low forward speed which makes the rudder rather inefficient.

324 M. Levadou and R. van’t Veer

Future enhancements are expected to overcome these limitations. Surge degree of freedom was neglected as well, for similar practical reason. The surge balance requires a dedicated implementation of added resistance, ship resistance and propulsion plant control, which is currently in development. The viscous roll damping in Pretti is currently based on a time-domain implementation of Ikeda’s method. For the bilge keel damping the water velocities at the bilge keel are assessed at each time instance and with a Keulegan-Carpenter number the drag on the bilgekeels are calculated. Lift en eddy damping follows Ikeda’s empirics. All simulations were performed in the same ‘relative’ sea state realisation. This means that is the wave component phase and relative amplitudes were kept identical when the significant sea state was varied. To obtain reliable statistics the simulations were performed for a duration of 3 hours. The computer time requires (CPU) is about half the simulation time. This means that a parametric study as presented in this paper is feasible in a design stage, although significant computer CPU is required for several days. A mesh of the C11 container ship as used in the simulations is presented in Fig. 12.

Fig. 12 C11 container ship mesh for time domain simulations

Model tests were carried out for two different hull shapes, the original C11 container ship and a modified one with the pram aft-body, denoted SIM Pram aft in Fig. 13. The significant roll amplitude compares very well between simulations and model tests for the lower sea states. In the higher sea states the simulations predict larger parametric roll motions then the model tests show. The trend is however well predicted and the significant wave height at which parametric roll starts agree well. The pram aft body might be beneficial for calm water resistance, but makes the design more sensitive for parametric roll. Striking is the fact that the original design shows less parametric roll in the higher sea states than the modified design, while this is opposite in the simulations. We consider this an effect of the not fully developed numerical models and the assumptions made in PRETTI. But,

Parametric Roll and Ship Design 325

most likely nonlinear hydrodynamics or more sophisticated nonlinear (viscous) damping is required.

Fig. 13 Significant roll amplitude from model experiments (EXP) and from nonlinear simulations (SIM) for the C11 and Pram modified hull

Following these results three other hull shape variations were investigated. The results are summarized in the Fig 14.

Fig. 14 Significant roll amplitude in Jonswap wave spectra for the C11 and 4 modified hull shapes

326 M. Levadou and R. van’t Veer

The numerical results are summarized in Table 3, which compares the threshold value with Dunwoody. A threshold significant roll amplitude of about 5 degrees was used. The comparison between Dunwoody and direct simulations is good, which is a striking conclusion. The dynamic GM variations in the simulations are to comparable but to some extent different from the GM variations in an assumed static wave due to the fact that the equilibrium buoyancy condition in waves is not obtained; heave and pitch motions will change the buoyancy of the ship. The numerical simulations show a stronger influence of the bow flare then the static simulations, which could be due to the mentioned reason; the position of the ship in waves will be different in simulations than in a static approach. Table 3 Comparison Dunwoody and Pretti simulations. Threshold wave height for parametric roll Dunwoody

Simulations

C11 original

4.3 m

4.2 m

Pram aft

3.2 m

3.5 m

V shape aft

5.3 m

5.1 m

Large bow flare

4.3 m

4.2 m

Small bow flare

4.3 m

4.6 m

5 Roll Stabilization In the previous chapter the influence of the hull form on the parametric roll threshold wave height has been demonstrated. One other way to influence the threshold wave height is to change the roll damping of the vessel. Usually vessels have a small potential damping. Therefore, adding appendages (bilge keels, fin stabilizers) or anti roll tanks can increase the roll damping drastically. The influence of bilge keels and active fin stabilizers on the threshold wave height is investigated by using Dunwoody’s method.

5.1 Bilge Keel The bilge keel damping is often associated with the energy dissipated by the drag forces of the bilge keel (Dallinga 1998). Within this concept the damping is proportional with the roll velocity amplitude. Equation (19) gives the increase of roll damping per roll velocity amplitude change.

Parametric Roll and Ship Design 327

BBK 4 2   2lBK hBK  CHF rBK  rBK CD , BK (19)  3  According to Ridjanonic (1962) the effective drag coefficient depends on the amplitude of the transverse flow and the bilge keel height, as per equation (20). hBK CD , BK  22.5  2.4 (20)  rBK CHF  Using these equations the bilge keel damping was calculated for several bilge keel heights. The contribution as a fraction of the total damping linearized for 10 deg roll amplitude is given in the first row of Table 4 for 5 knot speed. Table 4 Roll damping contribution from the bilge keel. Bilge keel height 0 cm

20 cm

40 cm

60 cm

Fraction

0

0.256

0.428

0.547

Total Damping [MNms/rad]

145

195

253

320

As can be seen the roll damping contribution of the bilge keel is very large. Using these fractions on the total roll damping determined from the roll decay tests, the damping for a variation of bilge keels can be determined. These values are indicated in the second row of Table 4. Using Dunwoody’s method the threshold wave height was determined for the different bilge keel heights. The results are given in Fig 15. As can be seen, a 40 cm bilge keel raises the threshold wave height by Hs = 1 m. Adding 20 cm raises the threshold by about 0.6 m.

Fig. 15 Influence of bilge keel height on threshold wave height

328 M. Levadou and R. van’t Veer

5.2 Active Fin Stabilizer The influence of active fin stabilizers can be estimated using the same approach. The roll damping (passive and active part) was estimated using the following equation (Dallinga 1993, 1998).  CL rFIN CHF CL  1 (21)  BFIN  2rFIN VS2 AFIN CFH  bC   act  2   pas VS Using these equations the damping of fin stabilizers was calculated for several fin stabilizers area. The damping linearized for a 10 deg roll amplitude is given in Table 5. Table 5 Roll damping contribution of fin stabilizers. Fin stabilizer area 0 m2

10 m2

21 m2

Fin Stab damping [MNms/rad]

0

87

173

Total Damping [MNms/rad]

253

340

426

Using Dunwoody’s method the threshold wave height was determined for the different fin stabilizers area. The results are given in Fig 16 for 5 knot speed. As can be seen a 10 m2 fin stabilizer raises the threshold wave height by Hs = 0.7 m. Adding 21 m2 fin stabilizer raises the threshold by about 1.0 m. The results show that the fin stabilizers give approximates the same increase in threshold wave height as bilge keels for 5 knot speed. However, for speeds higher the influence of the fin stabilizers will be higher. This is due to the fact that the fin stabilizer damping increasing with the square of the speed.

Fig. 16 Influence o fin stabilizers on threshold wave height

Parametric Roll and Ship Design 329

5.3 Results Summary The results of the simulations and model tests have shown that the C11 original hull form results in a lesser probability of parametric roll to occur than the pram aft body. The pram aft body is however beneficial with respect to calm water resistance. Using the results presented in the previous chapters an estimation can be made of the bilge keel height or fin stabilizer area needed on the pram aft body in order to have a comparable threshold wave height as the C11 hull form. This estimation is given in Table 6. The results show that in order to achieve the same threshold wave height the bilge keel must be twice as big or the pram aft body must be equipped with a 21 m2 fin stabilizer. These results are valid for 5 knots speed. For higher speeds the fin stabilizer will give more roll damping than the 0.80 m bilge keel. Table 6 Appendage size, bilge keel or fin stabilizer, needed for same threshold wave height. Appendage size Hull shape

BK height (m)

Fin stabilizer area (m2)

C11 original

0.40

-

Parm aft body

0.80

-

Pram aft body

0.40

21

6 Conclusions In the paper the influence of main dimensions, hull form and roll stabilization on the occurrence of parametric roll are discussed. A one degree of freedom motion method and nonlinear time domain simulations were used and validated with model tests. Regarding the results the following conclusion may be drawn. A relatively simple one degree of freedom motion method can be used in the preliminary stage of a design. An important factor on the results is the used roll damping. Empirical models for the roll damping, tuned with model tests, can be used in a preliminary stage. The method gives an idea of the threshold wave height for which parametric roll will start but not the actual roll angles associated with the rolling. Nonlinear time domain simulations can be used to determine the threshold wave height and to determine the roll angles associated with the parametric roll. Here also tuning of the roll damping is needed in order to get reliable results. The model tests and calculations on a C11 type container vessel have demonstrated that the aft body configuration has more influence on the occurrence of parametric roll than the bow flare. A V-shaped aft body is preferable to a pram type aft body.

330 M. Levadou and R. van’t Veer

Finally, the influence of bilge keel height on the occurrence of parametric roll has been shown. It has also been shown that active fin stabilizers can be used at low speed in order to increase the threshold wave height.

References Bailay PA, Price WG and Temarel P (1997) A unified mathematical model describing the manoeuvring of a ship travelling in a seaway, RINA Cummins WE (1962) The impulsive response function of ship motions, Schifftechnik, Band 7 Dallinga RP (1993) Hydromechanic Aspects of the Design of Fin stabilizers, RINA Spring Meet, Lond Dallinga RP, Blok JJ and Luth HR (1998) Excessive rolling of cruise ships in head and following waves, RINA Int Conf on ship Motions & Maneuverability, Lond Dunwoody AB (1989) Roll of a ship in astern Seas – Metacentric height spectra, J of Ship Res. 33: Nº 3: 221-228 Dunwoody AB (1989) Roll of a ship in astern Seas – Response to GM fluctuations, J of Ship Res. 33: Nº 4 : 284-290 Fossen TI and Smogeli ON (2004) Nonlinear Time-Domain Strip Theory Formulation for LowSpeed Manoeuvring and Station-Keeping, Modeling, Identification and Control. 25: 4 France WN, Levadou M, Treakle TW, Paulling JR, Michel RK and Moore C (2003) An Investigation of Head-Sea Parametric Rolling and its influence on Container Lashing Systems, Mar Technol 40: 1 Graff W and Heckscher E (1941) Widerstands und Stabilitäts versuche mit drei Fischdampfermodellen, Werft-Reederei-Hafen, 22 (also DTMB Translation 75, June 1942) Hogben N, Dacuhna NMC and Olliver GF (1986) Global Wave Statistics, BMT, Lond Ikeda Y, Himeno Y, Tanaka Y (1978) A prediction method for ship roll damping, Technical Report 00405, Univ of Osaka Kempf G (1938) Die Stabilitätsbeanspruchung der Schiffe durch Wellen und Schwingungen, Werft-Reederei-Hafen, 19 Levadou M, Palazzi L (2003) Assessment of Operational Risks of Parametric roll, World Maritime Technol Conf, San Francisco Luth HR and Dallinga RP (1998) Prediction of excessive rolling of cruise vessels in head waves and following waves, PRADS, The Hague Netherlands Naval Sea Systems Command: Ship Hull Characteristics Program User Manual version 4.3 (2003) Paulling JR and Rosenberg RM (1959) On Unstable Ship Motions Resulting from Nonlinear Coupling, J of Ship Res. 3: 1 Paulling JR (1961) The Transverse Stability of a Ship in a Longitudinal Seaway, J of Ship Res. 4:4 Ridjanovic M (1962) Drag coefficients of Flat Plates Oscillating Normally to their Planes, Schifftechnik, Band 9, Heft 45 Shin YS, Belenky VL, Paulling JR, Weems KM, Lin WM (2004) Criteria for Parametric Roll of Large Containerships in Longitudinal Seas, SNAME

Parametric Roll Resonance of a Large Passenger Ship in Dead Ship Condition in All Heading Angles Abdul Munif*, Yoshiho Ikeda*, Tomo Fujiwara**, Toru Katayama* *Osaka Prefecture University; ** National Maritime Research Institute Abstract Although it has been widely believed that a ship in dead ship condition has the largest roll motion at resonance condition, the authors found that, because of parametric roll resonance, a kind of ships has significant heavy rolling in beam waves with slightly smaller period of the half value of its natural roll period if the roll damping is small. In the present paper, model experiments to measure ship motions of a large passenger ship in waves with various heading angle are carried out to confirm the region of heading angles where large parametric rolling appears. The results demonstrate that parametric rolling appears at wide region of heading angles and disappears in certain heading angles.

1 Introduction As well known, occurrence of small parametric rolling of ships in beam seas has been pointed out by many researchers. In the previous paper proposed by Ikeda et al. (2005), however, the authors found experimentally that heavy roll motion with much larger angle than that in resonance appears for a large passenger ship with flat stern and large bow flare in heavy beam seas due to parametric rolling. The measured results demonstrated that the large parametric rolling appears only when the ship has no bilge keel or smaller one. As wave height increases, the parametric rolling suddenly appears at higher wave height than a certain value. The authors also did a simulation of roll motion, and confirmed that similar parametric rolling appears for the ship in beam waves (Munif et al. 2005). In the present study, some additional experimental works to clarify effects of wave height, size of bilge keels and heading angles of waves on occurrence of large parametric rolling are carried out to reveal the characteristics of the parametric rolling.

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_18, © Springer Science+Business Media B.V. 2011

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332 A. Munif et al.

2 Experimental Setup The ship used in the experiments is the 110,000GT passenger ship designed by Fincantieri for an international cooperated research on damage stability of large passenger ships in IMO. The body plan and the principal particulars are shown in Fig. 1 and Table 1, respectively. The bilge keels designed for the ship are divided into two parts, short forward one and long aft one. The roll damping can be changed by selecting these bilge keels.

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

Breadth

36 m

0.287 m

Draft

8.4 m

0.067 m 8

Displacement

5.20 × 10 N

264.67 N

GM

1.579 m

0.0126 m

T roll

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

In the previous study, a scale-model of the large passenger ship was located in transverse direction in the towing tank of Osaka Prefecture University, and the ship motions, roll, heave, pitch, sway and drift motions, were measured in regular beam waves. In the experiment, yaw and surge motions were fixed.

Parametric Roll Resonance of a Large Passenger Ship 333

In the present study, the same model is located in regular waves for freely yawing conditions or various fixed heading angles for incident waves. Only transverse motion of the model is fixed by a guide. This means that drift motion in incident-wave direction is free but the motion perpendicular to it is fixed. The experimental conditions are shown in Table 2. Table 2 Experimental conditions -

Full scale

Model

Wave period

6.7 – 24.6 sec

0.6 – 2.2 sec

Wave length

70 – 944 m

0.56 – 7.55 m

1.25 – 10 m

0.01 – 0.08 m

Wave height

without , full ( front + aft ) , front , aft Bilge keel

front : s.s. 5.25 – 6.0 aft : s.s. 3.0 – 5.0

3 Experimental Results As mentioned before, in the previous paper, the authors experimentally showed that the effect of wave height on parametric rolling in regular beam waves. To confirm the results in wider wave height region, parametric rolling of the ship without bilge keels is measured for various wave heights. The maximum amplitude of the parametric rolling at each wave height is shown in Fig. 2. The results demonstrate that the parametric rolling appears just over 30mm of wave height, rapidly increases with wave height, and reaches the maximum amplitude that is about 27 degree. It should be noted that the roll amplitude does not proportionately increase with increasing the wave height but seems to saturate to the maximum one. In Fig. 3, effect of area of bilge keels on maximum amplitude of parametric rolling is shown. The results demonstrate that the parametric rolling significantly depend on the area of bilge keels. It is safely said that large bilge keels can completely erase parametric rolling in beam seas. To investigate the effect of heading angle on the parametric rolling of the dead ship, the models are released in head wave condition (χ=180°) or following wave condition (χ=0°) in regular waves with 0.04m of wave height and 0.95 seconds of wave period which is the period when the large parametric rolling in beam seas appears. The time histories of the heave, pitch, roll and yaw motions are measured and showed in Figs. 4 and 5. The model has no bilge keels in these measurements.

334 A. Munif et al.

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 time histories of roll motion in these figures show that the period of the roll motion is twice to the heave and pitch motions. The results demonstrate that the parametric rolling occurs in wide heading angles as well as in beam seas. 30 25

Φ 0 (deg)

20 15 10 5 0

0

20

40

H w (mm)

60

80

100

Fig. 2 Effect of wave height on maximum amplitude of parametric rolling of the ship in regular beam seas. 30 25

Φ 0 (deg)

20 15 10 5 0

0

0.2

0.4

0.6

0.8

1

Area of Bilge Keel / Area of Designed Full Bilge Keel

Fig. 3 Effect of area of bilge keels on maximum amplitude of parametric rolling in regular beam seas.

Parametric Roll Resonance of a Large Passenger Ship 335

Fig. 4 Time histories of motions of the ship without bilge keel released from head sea condition in regular waves at Tw=0.95sec and Hw= 0.04m.

336 A. Munif et al.

Fig. 5 Time histories of motions of the ship without bilge keel released from following wave condition in regular waves at Tw= 0.95sec and Hw= 0.04m.

Parametric Roll Resonance of a Large Passenger Ship 337

Since it may take time to lead to the parametric rolling and reach its steady condition, a measurement of roll motion at some fixed heading angle; 0, 90, 180 degree are carried out. And using the results of above-mentioned measurement and of previous ones shown in Figs. 4 and 5, the roll amplitude of the parametric rolling for each heading angle is plotted in a polar diagram as shown in Fig. 6. The results show that amplitudes of the parametric rolling are significant in following and head seas as well as beam seas when no bilge keel is attached.

Fig. 6 Effect of wave direction on amplitude of parametric rolling of the ship without bilge keel in regular waves. 2 1.5 1

GZ(m)

0.5 0

-0.5 0

10

20

30

40

50

60

-1

-1.5 -2 -2.5

Still water Wave trough wave crest

-3

heel angle (deg) Fig. 7 Variation of GZ curves in head waves of λ/L=1.5 and Hw/λ=0.033.

338 A. Munif et al.

2 1.5 1 GZ(m)

0.5 0

0

10

20

30

40

50

- 0.5 -1 - 1.5 -2

still water wave trough wave crest φ (deg)

Fig. 8 Variation of GZ curves in beam waves of λ/L= 0.47 and Hw/λ=0.04.

Figs. 7 and 8 show variation of GZ-curves of the ship on the basis of FroudeKrylov assumption in heading and beam seas, respectively. We can see that the GZ-curves significantly vary between wave crest and wave trough in head and beam waves. These large variations of the GZ-curve create the large amplitude of parametric rolling. To investigate the effect of roll damping on occurrence of parametric rolling in various heading angles, motions of the model with bilge keels are measured in regular waves (Hw = 0.08m, Tw = 0.95 sec). In Fig 9 the time histories of motions of the ship with smaller front bilge keels released from following wave condition in the regular waves are shown. Smaller parametric rolling occurs in all heading angles of waves. When larger aft bilge keels or full bilge keels are attached, no parametric rolling appears as shown in Figs. 10 and 11.

Parametric Roll Resonance of a Large Passenger Ship 339

Fig. 9 Time histories of motions of the ship with short (front) bilge keels released from following wave condition at Tw=0.95sec and Hw= 0.08m.

340 A. Munif et al.

Fig. 10 Time histories of motions of the ship with long (aft) bilge keels released from head wave condition at Tw= 0.95sec and Hw= 0.08m.

Parametric Roll Resonance of a Large Passenger Ship 341

Fig. 11 Time histories of motions of the ship with full (front and aft) bilge keels released from head wave condition at Tw=0.95sec and Hw= 0.08m.

342 A. Munif et al.

To search the possibility of occurrence of parametric rolling of the ship with bilge keels in head waves, measurements of ship motions of the model with full bilge keels in fixed yaw condition are carried out in regular head waves with 0.08m of wave height and various wave periods. The results are shown in Fig. 12. The results suggest that parametric rolling with about 10 degrees can occur for the ship with full bilge keels in head waves at 1.2 second of wave period, which is longer by 26% than that when parametric rolling in beam seas occurs. In Fig. 13, the time histories of motions of the case that the maximum parametric rolling occurs are shown. It should be noted it takes long time to develop parametric rolling. In Fig. 14, the effect of heading angle on parametric rolling of the ship with full bilge keels in head waves is shown in a polar diagram. We can see that in heading angles of  20 degrees in head and following waves parametric rolling occurs even if size of bilge keels is enough to erase it in beam seas.

12 Tw = wave period

10

Troll = roll natural period

φ 0 (deg)

8 6 4 2 0

0

0.5 Tw / Troll

1

Fig. 12 Parametric rolling of the ship with full bilge keels in regular head waves with Tw= 1.2sec and Hw= 0.08m.

Parametric Roll Resonance of a Large Passenger Ship 343 wave hight (mm)

50 30 10 - 10 0

10

20

30

40

50

60

70

80

50

60

70

80

50

60

70

80

50

60

70

80

- 30 - 50

time (sec)

heave (mm)

030

0 0

10

20

30

-030

40

time (sec)

pitch (deg)

4 2 0 0

10

20

30

40

-2 -4

time (sec)

roll (deg)

30 20 10 0 - 10

0

10

20

30

40

- 20 - 30

time (sec)

Fig. 13 Time histories of motions of the ship with full bilge keels released from head wave condition in regular waves with Tw= 1.2sec and Hw= 0.08m.

344 A. Munif et al.

Fig. 14 Effect of wave direction on amplitudes of rolling and its period of the ship with full bilge keels in regular waves.

In Fig. 15, difference of wave periods for which parametric rolling appears in beam and head waves. The ship has no bilge keels and wave height is 0.04m. The results suggest that parametric rolling appears in different wave periods in beam and head waves. This may be because of differences of drift speed and amplitude of stability variation in beam and head waves. 30 Hw = 0.04m 180deg head sea 90deg beam sea

20

without Bilge Keel

φ 0 (deg)

25

15 10 5 0

0

0.5

λ / Lpp

1

1.5

Fig. 15 Difference of wave periods for which parametric rolling appears in beam and head waves.

Parametric Roll Resonance of a Large Passenger Ship 345

4 Conclusion Following the previous paper of the authors in which large parametric rolling in beam seas is confirmed for a large passenger ship with a flat and shallow sternbottom and large flare in bow in no bilge keel condition, parametric rolling in all heading angles of the same ship is experimentally investigated. Following conclusions are obtained. 1. When the ship has no bilge keels, parametric rolling occurs in wide heading angles including head and following seas as well as beam seas. Large variation of GZ value in waves due to a flat and shallow stern-bottom and large flare in bow induces the parametric rolling. 2. The effect of bilge keels on occurrence of large parametric rolling is significant. Large bilge keels can completely erase parametric rolling of the large passenger ship in dead ship condition in any heading angles of waves except head and following waves. 3. Parametric rolling of the ship in head seas can occur even if large bilge keels enough to erase it in beam seas are attached. 4. Wave periods when parametric rolling occurs in beam and head waves are different.

Acknowledgements This work was a part of the first author’s post-doctoral research supported by Grand-in-Aid funded by the Japanese Society for the Promotion of Science (JSPS). Just after the present study, Dr. A. Munif was died on the 9th, February 2006 because of sickness. The other authors express their deepest regret for him and his family.

References Ikeda Y, Munif A, Katayama T et al (2005) 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. Munif A, Katayama T, Ikeda Y (2005) Numerical Prediction of Parametric Rolling of a Large Passenger Ship in Beam Sea. Conference Proc. of the Japan Society of Naval Architects and Ocean Engineers, No.1.

Parametric Rolling of Ships – Then and Now J. Randolph Paulling Professor Emeritus, University of California

Abstract Modern research on parametric rolling of ships was first conducted in Germany in the late 1930s. This work was initiated in an effort to explain the capsizing of some small ships such as coasters and fishing vessels in severe following seas. The work included experiments with models in open water as well as numerical and theoretical computations. The phenomenon was thought to be of concern principally in following seas and for small, low freeboard ships. In the 1990s, however, there were reports of containerships and even some cruise ships experiencing heavy rolling in head seas. These were ships having a hull form characterized by great flare forward and wide flaring stern sections, features that are known to lead to significant stability variations in waves. The APL CHINA casualty in October 1998 focused attention on parametric rolling in head seas. The investigation of this casualty included theoretical computations of GZ variations, model experiments and numerical simulations as well as meteorological studies of the wind and sea conditions prevailing at the time. Results of the investigation received wide dissemination in the technical press. Since the APL CHINA casualty and other similar incidents, much theoretical and experimental work has been focused on head seas parametric roll. IMO and many of the Class Societies now have recommendations to designers and masters for avoiding head seas parametric rolling situations.

1 Introduction The phenomenon of parametric roll has been recognized by naval architects for more than fifty years. It may be described as a spontaneous rolling motion of the ship moving in head or following seas that is not directly excited by the waves but comes about as a result of a dynamic instability of the motion. In pure head or following seas the transverse symmetry of the ship would imply that no waveinduced roll exciting moment should be present. Nevertheless, for certain frequencies of wave encounter, it is found that a small initial disturbance in roll can trigger an oscillatory rolling that can grow to appreciable amplitude after only a few cycles. M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_19, © Springer Science+Business Media B.V. 2011

347

348 J.R. Paulling

This is explained in terms of a dynamic instability or bifurcation in the motion characteristics. In 1863, William Froude identified a ship roll-heave coupling phenomenon akin to parametric roll consisting of heave oscillations at the heave natural frequency excited by the small roll induced variations in the buoyant force which occur at twice the roll frequency. The occurrence of parametric roll is directly related to variations in the transverse stability of the ship as it moves through head or following waves. As the ship encounters successive waves, the geometry of the immersed portion of the hull varies as the waves move along the ship length. The underwater geometry changes with time due to the pitch-heave motion combined with the wave profile. This results in a time varying transverse stability as measured by both GM for small angles of heel and the GZ curve at large angles. Figure 1 contains such righting arm curves for a general cargo ship of 1960s vintage and a modern post Panamax containership, both in waves of length equal to ship length and height equal to L/20. The general character of the time varying curves is illustrated here consisting of diminished stability when the wave crest is amidships and enhanced stability when a trough is amidships. For intermediate positions of the waves the stability values vary accordingly.

2 A Simple Dynamic Model of Parametric Roll The dynamic rolling motion of a ship undergoing these stability variations in head or following seas is analogous to that of a spring-mass system in which the spring constant varies sinusoidally in time. For the ship the frequency of these variations equals the frequency of wave encounter. Let us now examine the small amplitude roll dynamics of a ship with periodic stability variations using a simplified model as follows. As our ship moves through head or following seas the time-varying GM may be expressed as Equation (1).

GM (t )  GM o (1  C cos  t )

(1)

Here, GMo = still water GM, C = fractional variation of GM due to waves, heave and/or pitch,  = encounter frequency. The equation of motion for uncoupled roll motion without excitation is now given by Equation (2).

Ix

d 2   (GM o  CGM o cos t )  0 dt 2

(2)

Parametric Rolling of Ships – Then and Now 349 C4-S-57a IN L/20 WAVES

1,6 1,4 1,2 1

SW

GZ - M

0,8

CRST MID TRGH MID

0,6 0,4 0,2 0 -0,2

0

10

20

30

40

50

60

70

80

-0,4

HEEL - DEG

C11 CONTAINERSHIP IN L/20 WAVES

2,5 2 1,5

GZ - M

1

SW CRST

0,5

TRGH

0 -0,5

0

10

20

30

40

50

60

70

80

-1 -1,5

HEEL - DEG

Fig. 1 Righting arm curves for two ships in calm water and following seas: C4 – AMERICAN CHALLENGER Cargo Ship of 1960s vintage and C11 Post Panamax Containership (Body plans not to scale).

350 J.R. Paulling

Here, = angle of roll, Δ = ship displacement, Ix = mass moment of inertia in roll, including added mass effect. Now divide both sides of (2) by Ix and make the change of variable, 2 d2 2 d    t for which 2   . dt d 2 GM o 2 We note that  n  , Ix where  n  natural frequency of roll.

We define,



GM o  n2  2,  2Ix 



C GM o  n2  C . I x 2 2

The equation of roll motion now becomes,

d 2  (   cos  )  0 . d 2

(3)

This ordinary differential equation with sinusoidally varying spring constant is known as the Mathieu equation and the behavior of its solutions have been determined. Figure 2, known as the Ince-Strutt diagram illustrates the stability of solutions for the Mathieu equation. In this diagram the shaded regions represent stable solutions to the equation and unshaded regions correspond to unstable solutions. Thus, if the two parameters (δ,ε) for the system lie in a stable region, an arbitrarily small initial disturbance will die out with increasing time, while if they lie in an unstable region, the disturbance will grow with time. We see that  is equal to the square of the ratio of the natural frequency of roll to the frequency of variation of GM and  is proportional to the fractional change in GM. The first unstable region is centered on a value =1/4 or a ratio of natural frequency to frequency of GM variation of ½. If the frequency of GM variation does not exactly satisfy this value, unstable motion can still occur for a sufficiently large value of C, i.e., if the amplitude of the variation in GM is sufficiently large. The effect of linear damping is to merely raise the threshold value of C at a given frequency of variation, . The unstable motion will still take place if C is sufficiently large and, in general, will grow without bound. In order for a limit on the ultimate amplitude to exist, there must be nonlinear damping in the system of which quadratic roll damping is an example. From the foregoing simplified analysis we can expect that, if the ship encounters regular head or following seas at a frequency near one-half the natural frequency of roll, a small disturbance in roll will grow to appreciable amplitude depending on the amplitude of the stability variation and the roll damping. In real situations, the initial disturbance is almost always present and supplied by some external effect such as wind or oblique wave components.

Parametric Rolling of Ships – Then and Now 351

Fig. 2 Ince-Strutt diagram illustrating stability of solutions of the Mathieu Equation.

3 Early Parametric Roll Research Much of the earliest work on parametric roll was conducted in Germany starting in the late 1930s and focused on smaller working ships such as fishing boats, seagoing tugs and small coastal cargo craft which had experienced a number of casualties. Some of the research involved experiments with free running models which demonstrated the phenomenon quite convincingly. The reasons for concentrating on smaller ships, aside from their casualty records, were two in number. First, smaller ships are more likely than large ships to encounter waves that are high and steep in relation to the ship dimensions. Second, they generally operate at Froude numbers that are more likely to result in a frequency of encounter within the critical range for the ship of marginal stability. Such ships are, of course, the ones in greatest danger of capsize. Further, it was felt that the transient behavior of the roll motion was rather gradual, i.e., a large number of cycles would be required in order for the rolling motion to grow to appreciable amplitude. This was due in part to observations during the aforementioned model tests and in part to some early numerical simulations of the phenomenon which showed a slow growth of amplitude of the roll motion. This slow transient growth was interpreted to mean that in order for the rolling motion to grow to appreciable amplitude, a rather large number of nearly regular waves would have to be encountered.

352 J.R. Paulling

Fig.3 Unidirectional random waves observed from moving and stationary points.

Waves in the real ocean, however, are random in character although in a system of random waves there will, from time to time, be found groups of several waves that appear to be almost regular. Furthermore, the waves as observed from a moving point appear to be somewhat different from the waves at a fixed point. To a moving ship, therefore, following waves appear to have distinctly different characteristics compared to head seas. This is a consequence of the dispersion relation, i.e., the dependence of wave velocity on wave length. When observed from a point moving in the same direction as the waves, as in following seas, the wave spectrum is compressed into a narrower band of frequencies than that for a fixed point, while head seas result in a spreading of the spectrum over a wider band of frequencies. The resulting appearance of the wave time history is shown in Figure 3. Here, the upper graph contains the wave profile observed from a point moving in the same direction as the waves, the center graph is the profile observed from a fixed point and the lower graph corresponds to a point moving opposite to the wave motion.

4 The APL CHINA Casualty In the early 1990s several containerships reported the occurrence of heavy rolling in head seas and the description of the incidents indicated the possibility of parametric rolling. Some of the reports indicated that the rolling motion built very quickly, during the encounter of three or four waves not several tens of waves as would have been expected. These containerships had hull forms characterized by great flare forward and wide flaring stern sections. The reason for these form features was a desire to obtain more container space on deck while keeping a fine underwater form to minimize resistance at the high operating speeds of the ships. These form features, however, lead to large stability changes as the ship moves through waves from the crest amidships to trough amidships positions. Furthermore, the typical metacentric heights at which these ships operate put their natural roll frequencies into the range of one-half the head sea encounter frequencies for wave lengths approximately equal

Parametric Rolling of Ships – Then and Now 353

to ship length. These are typically the waves causing the most pronounced stability variations. The effect on stability of the flared form can be seen in Figure 1. At the smaller angles of heel, the righting arm curves, trough versus crest amidships, diverge much more strongly for the C11 containership than for the older, more wall-sided dry cargo ship of traditional form. This implies a larger value for C, the coefficient of variation of GM in Equation (1) and would lead to increased susceptibility to occurrence of parametric roll, all other things being equal.

6 TH International Ship Stability W ork sh op. A n Investigation of H ea d-Sea Param etric R olling and its Influence on C ontain er L ashing System s, O ctob er 200 2.

Fig. 4 APL CHINA container damage.

The APL CHINA casualty in October 1998 focused worldwide attention on head seas parametric roll. While eastbound in the North Pacific Ocean, the ship found itself in a rapidly moving weather system that was formed by the convergence of two low pressure systems and, despite weather routing advice, was unable to avoid the rapidly changing situation. The resulting combined system was described as an “explosively intensifying low” or meteorological ‘bomb” and referred to by some as a Pacific version of the “perfect storm”. At the height of the storm, the ship’s deck log recorded estimated wind of Beaufort force 11 and sea state 9. A weather and sea state hindcast performed after the accident gave peak sustained winds of 30.8 m/s or 60 knots and a significant wave height of 14.9 m During the height of the storm, the master attempted to maintain a heading into the seas at a speed just sufficient to retain control. The ship reported rolls of over 40 degrees and violent pitching in the very heavy and confused seas. As a result of the violent motions combined with boarding seas, heavy damage to- and loss of a large number of containers was sustained and this is illustrated in Figure 4. In order to determine the causes of this casualty, extensive studies including weather analyses, model experiments, numerical simulations, and theoretical analyses were undertaken. At the conclusion, it was felt that head seas parametric roll was clearly established as the major cause. One significant outcome of this and related work is the development by classification societies as well as IMO of rules and guidelines for the avoidance of parametric roll situations both in the design and operation of ships. Much more fundamental research on parametric roll has been conducted since the APL CHINA studies. Many model tests have been conducted, simulation codes have been developed and exercised, and theoretical work has been conducted on

354 J.R. Paulling

the probabilistic nature of nonlinear phenomena. A considerable body of literature on the subject exists, much of it the product of the STAB symposia. Rather than attempting to summarize it we will instead take a look at two specific side issues, one concerning the nature of the transient roll response in regular waves and wave groups and the second concerning the difference in behavior of the stability variations in head seas in contrast to following seas.

5 A Somewhat More Sophisticated Model Let us consider a slightly modified form of Equation (2) as follows,

d 2 d  B( )   GZ ( , t )  0 (4) 2 dt dt d Here, B ( ) is a general roll damping which may include linear and nonlinear dt (quadratic) terms. GZ ( , t ) is the time varying restoring moment function. Ix

The most suitable method of solving equation (4) is stepwise numerical integration and a number of suitable methods are available. The forth order Runge-Kutta method was used in the present examples. The function GZ ( , t ) may be visualized as a tabulated family of curves similar to those in Figure 1 computed for successive closely spaced positions of the wave along the ship length. The damping term may be estimated from data found in the literature and many ship hydrostatics codes are available for constructing the family of GZ curves. A simple interpolation scheme is used for finding the instantaneous value of GZ for the angle of heel and wave position at each time step of the integration. Having coded this numerical integrator, it is easy to play “what if” games that may lead to understanding of some of the basic features of parametric roll response. Two examples are presented here, the first consisting of the response time history for the ship in regular head seas where the computation is started with different initial conditions in each case. The second looks at the response in head seas consisting of a series of repeated wave groups. The groups are formed by modulating the fundamental sinusoidal wave train by a sinusoidal envelope function of lower frequency chosen to produce groups somewhat resembling those encountered in a random seaway. The C11 at its designed draft having a GM of 0.05B is used as the example in both of these computations. The family of GZ curves was constructed for 20 equally spaced positions of the wave crest along the ship length and the nonlinear roll damping was estimated using the method embedded in the US Navy’s SMP code. The natural undamped roll period of the ship was estimated to be about 23 seconds assuming a mass radius of gyration in roll of 0.4B. In head seas of length

Parametric Rolling of Ships – Then and Now 355

equal to the ship length, the critical frequency ratio of ½ occurs at a speed of about 5.5 knots. In constructing the wave groups for the second set of computations, we first examine Figure 3 for some impression of the behavior of random seas. The middle graph of Figure 3 corresponds to observation from a fixed point in space and shows wave groups at about 20, 400 and 700 seconds. Each group consists of four to six waves. At the higher speed of the lower graph, there are groups at about 10, 320, 500 and 650 seconds. The latter groups appear to have fewer component waves and persist for shorter time periods. For the present example, we have constructed a wave system in which each group contains five principal waves formed by multiplying the fundamental ship length sine wave whose period is about 12.9 seconds by a modulating sine wave with period of 129.5 seconds. This is found to produce individual wave groups having an appearance similar to that which might be found in a real wave system.

6 Initial Condition Sensitivity Figure 5 presents results in regular waves for an initial roll of 0.1, 1.0 and 5.0 degrees. As one would expect, this demonstrates that the transient build up of the motion to steady state is strongly dependent on the initial disturbance. Reported cases of unexpected head seas parametric roll have indicated that the motion built up rapidly over the occurrence of five or six extreme rolls and then subsided. In heavy head seas some component waves from directions other than dead ahead will always exist and these could very likely set up roll motions of five degrees or so that would then trigger the rapid build up of parametric response.

Fig. 5 Sensitivity of transient response to initial disturbance for initial roll angle = 0.1, 1.0, 5.0 deg.

356 J.R. Paulling

7 Wave Group Response The transient response in group waves appears somewhat more sluggish than in regular waves with initial conditions of both one and five degrees as shown in Figure 6. There is some sensitivity to the time of application of the initial roll disturbance with respect to the peaks and nodes of the envelope, but computations in which the time of application was varied showed it to be of secondary importance to the magnitude of the initial roll angle.

Fig. 6 Response of C11 to wave groups, head seas, initial roll angle = 1.0 and 5.0 degrees.

8 Sensitivity of GZ Values to Forward Speed Several of the hydrostatic stability computer codes in common use by naval architects today are capable of computing the GZ curves for the ship assumed poised in static equilibrium on the wave. It has been shown in some experiments by Paulling (1961) that this assumption of static equilibrium is usually satisfactory for moderate speeds in following seas. In these experiments, a model was towed at a fixed angle of heel using an apparatus that allowed the model freedom in pitch and heave. As the model was towed on a course at right angles to the wave crests the righting moment was continuously measured and recorded. Results from experiments conducted at various model speeds in following seas revealed very little variation with model speeds in the range of the normal speeds for seagoing ships. These experiments were carried out on a family of models representing the DTMB Series 60 constructed with variations in beam and freeboard. The experiments were conducted in following seas only and did not explore the possible effects that might be associated with the greater pitch-heave motions to be expected when the ship operates in head seas. Examples of results obtained in following sea experiments compared with hand computations are shown in Figure 7.

Parametric Rolling of Ships – Then and Now 357

The computations assumed static equilibrium of the ship on wave and the agreement is seen to be good for the following seas case. In head seas, the assumption of static equilibrium of pitch and heave is probably not as valid an approximation as it is in following seas. This is especially true at higher speeds for which the encounter frequencies may approach resonance in the pitch-heave modes.

Fig. 7 Model measurements of GZ in following seas. Series 60 Cb=0.60, Varied freeboard.

In order to test the sensitivity of the GZ curve to forward speed, an approximate computation procedure was adopted which would simulate the model experiments. It was assumed that the pitch-heave motion in head seas of the heeled ship would differ only slightly from the corresponding motion of the upright ship. With this assumption, these motions could be computed using a standard linear ship motions code. The pitch-heave motion time history so obtained was then input to the wave stability computations to give a closer approximation to the exact dynamic attitude of the ship in waves than given by the usual static equilibrium assumption. The results could then be regarded as an approximation to the GZ for the dynamically pitching and heaving ship. The dynamic pressure distribution on the ship due to the motions would be neglected as would the effect of heel on the pitch-heave motion. Since the Froude-Krylov pressure terms are usually dominant in determining the forces on the ship in waves, this procedure can be thought of as providing a first order correction for the exact dynamic attitude of the ship in the waves. An example of the ship’s changing attitude with passage of the waves is shown in Figure 8.

358 J.R. Paulling

Fig. 8 C11 Containership in L/20 head seas at 18 knots.

Figure 9 contains results of these computations in the form of maximum and minimum GZ curves. These correspond to a wave trough and a crest near amidships, for the ship moving at 18 knots in L/20 head and following seas. Also shown are GZ curves for an assumed static pitch-heave attitude of the ship which simulates following seas. The dynamic motion curves are labeled “F” for following and “H” for head seas. From these results we conclude that the effect of pitch-heave dynamics is to reduce somewhat the variation of the GZ curves about the still water curve. Further, we see that the assumed static pitch-heave attitude is quite close to the dynamic attitude in following seas.

9 Conclusions A very brief background is given of some of the practical implications of parametric roll for ships in following and head seas. Early work on the phenomenon had focused almost entirely on rolling and capsize in following seas but the APL CHINA casualty brought out quite dramatically the possibility and consequences of head seas parametric roll. Much research has come about as a result and we now find that regulatory bodies and shipowners are beginning to take steps to avoid a repetition of the incident both in the ship design process and in the operation of ships.

Parametric Rolling of Ships – Then and Now 359

C11 CONTAINERSHIP GM=0.05B L/20 WAVES

2,5 2 1,5

GZ - M

1 0,5 0

-0,5

0

10

20

30

40

50

60

70

80

-1 -1,5

HEEL - DEG Fig. 9 Max – min GZ curves for C11 containership – dynamic pitch-heave attitude at 18 knots.

Acknowledgement The work reported upon here has been supported over a period of many years principally by the US Coast Guard, the American Bureau of Shipping and the Society of Naval Architects and Marine Engineers

References Arndt B, Rodin S (1958) Stabilität bei vor-und-achterlichem seegang. Schiffstechnik. V 5: 29, Nov 192-199. DeKat JO, Paulling JR (1989) The simulation of ship motions and capsizing in severe seas. Transactions. The Soc of Nav Archit and Mar Eng V 97 France WN, Levadou M, Treakle TW, Paulling JR, Michel RK, Moore C (2003) An investigation of head-sea parametric rolling and its influence on Container Lashing Systems. Mar Technol. V 40:1 Jan 1:19 Froude W (1863) Remarks on Mr. Scott-Russell’s paper on rolling. Transactions. The Inst of Nav Archit Graff W, Heckscher E (1942) Widerstands und Stabilitäts versuche mit drei Fischdampfermodellen. Werft-Reederei-Hafen V 22 1941:115-120 (also DTMB Translation 75 June) Grim O (1952) Stabilität und sicherheit im Seegang. Schiffstechnik. V 1:10-21 IMO (1995) Guidance to the master for avoiding dangerous situations in following and quartering seas. MSC Circ. 707 dated 19 Oct Kempf G (1938) Die stabilitätsbeanspruchung der schiffe durch Wellen und Schwingungen. Werft-Reederei-Hafen. V 19:200-202 Kerwin J E (1955) Notes on rolling in longitudinal waves. Int Shipbuild Prog V2:597-614 Oakley OH, Paulling JR, Wood PD (1974) Ship motions and capsizing in Astern Seas. Proc Tenth ONR Symp on Nav Hydrodyn ONR ACR 204

360 J.R. Paulling Paulling JR, Rosenberg R M (1959) On Unstable ship motions resulting from nonlinear coupling. J Ship Res. V 3:1 Paulling JR (1961) The transverse stability of a ship in a longitudinal Seaway. J of Ship Res. V 4:4 Shin YS, Belenky VL, Paulling JR, Weems KM, Lin WM (2004) Criteria for parametric roll of large containerships in longitudinal seas. Transactions. SNAME. V 112:14-47 Tikka K, Paulling JR (1990) Prediction of critical wave conditions for extreme vessel response in random seas. Proc Stab90 Naples Wendel K (1954) Stabilitätseinbussen im Seegang und durch Koksdeckslast. Hansa 2016-2022 Wendel K (1960) Safety from capsizing. Fishing boats of the world 2 - J O Traung Ed. Fishing News Books Ltd. London 496-504

4 Broaching-to

Parallels of Yaw and Roll Dynamics of Ships in Astern Seas and the Effect of Nonlinear Surging K.J. Spyrou School of Naval Architecture and Marine Engineering, National Technical University of Athens, 9 Iroon Polytechneiou, Zographos, Athens 15780, Greece

Abstract In the first part of this paper we consider in parallel the roll and the yaw dynamics in astern seas and we point out a number of interesting fundamental dynamical analogies between the two modes. In the second part we focus on the roll dynamics, especially the parametric and the pure-loss mechanisms; however taking into account nonlinear surging effects. Analytical solution for the nonlinear surge motion is proposed. Moreover, characteristic graphs showing the quantitative effect of surging on capsize tendency are presented. Other aspects considered are, the effect of restoring modulation based on two frequencies and the dynamic effect of an initially hardening restoring.

1 Introduction As is well-known, fluctuations of the roll righting-arm in large following waves can result in ship roll instability and capsize according to the pure-loss or the parametric mechanism (Grim 1952). Fluctuations of a similar nature, concerning the motion’s stiffness term, may take also place in actively controlled yaw, originating from the combined effect of rudder control with the wave induced yaw moment. This could give rise to course instability resulting in deviation from the desired heading and possibly in broaching-to (Spyrou 1996). Consider a ship operating in long following sinusoidal waves. In order to avoid coupling complications let us assume further that, due to high natural frequencies in heave and in pitch compared to the encounter frequency, the ship can maintain a state of quasi-static equilibrium on the vertical plane. If the waves are relatively steep, the geometry of the submerged part of the hull will vary according to ship’s position on the wave. This is likely to be reflected in the roll righting-arm, with a reduced or even negative roll restoring arising when the middle of the ship is near to a wave crest. If roll restoring remains negative for sufficient time, so that heel finds the time to develop unopposed from some small initial angle up to levels M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_20, © Springer Science+Business Media B.V. 2011

363

364 K.J. Spyrou

well beyond the vanishing angle, then capsize due to the so-called pure-loss of stability mechanism will be realised (Paulling 1961). In this case the magnitude of roll damping affects insignificantly ship survivability. Capsize can occur of course also in a typical parametric resonance fashion and here the magnitude of damping should be a much more critical factor (Kerwin 1955). Practically, the variation of restoring must be however quite intensive, so that the large-amplitude roll motion could build-up within a small number of wave cycles (Spyrou 2000a). The onset of yaw instability in a similar environment represents a slightly more complex process; because yaw is always coupled with sway and also the control law of the rudder bears a serious influence on the dynamics. Unlike with roll, in the absence of active rudder control no restoring yaw moment can be produced in still water. In waves however, the movement of the rudder is intended to bring the ship back to her course. If waves with a length equal to the ship length or longer meet the ship from behind, they will create a yaw moment which will be dependent upon the angle between the direction of wave propagation and the ship’s heading. This wave yaw moment works as a positive restoring component when the ship passes from a wave crest (stabilizing effect). The opposite will be happening in the vicinity of a trough because the wave will tend to bring the ship vertical to the direction of wave propagation. As a result, the comparative strengths of the rudder and wave yaw moments will give rise to a restoring fluctuation of a certain amplitude which may sometimes have the potential to destabilize the horizontalplane motion of the ship. The commonality of the underlying fundamental dynamics of yaw and roll becomes therefore prevalent. Our first objective in this paper is to identify the correspondence between key yaw and roll motion parameters from the perspective of these Mathieu-type phenomena. As is well known, when the waves are large, the nonlinearity of surge during “following-sea” operation cannot be neglected (Kan 1990). A manifestation of this nonlinearity is a virtual rescaling of time as the ship is spending longer time on the crests than on the troughs of the waves. Despite the significance of this mechanism for the yaw and roll motions, there has been no earlier assessment of its effect for ship survivability and safety. The second objective therefore is to shed some light on the effect of surging on well-known capsize mechanisms.

2 A Simple Model of Yaw Motion in Astern Seas Consider the linear differential equations of sway and yaw (Clarke et al. 1983), with the addition of wave excitation terms at their right-hand side: Sway:

m  Yv  v  Yv v  mxG  Yr  r  m  Yr  r   Y  Ywave 

(1)

Parallels of Yaw and Roll Dynamics of Ships in Astern Seas 365

 ) Yaw: mxG  N v  v  N v v  I z  N r  r  mxG  N r  r   N    N (wave

(2)

In the above v, r  are respectively sway velocity and yaw angular velocity,  is the rudder angle, m is ship mass and xG is the longitudinal position of ship’s centre of gravity; Yv, Yr , N v , N r are acceleration coefficients (added masses/

moments of inertia) and Yv, Yr , N v , N r , Y are velocity coefficients (hydrodynamic damping terms). Y(wave) , N (wave) are respectively the wave’s sway force and yaw

moment. The prime indicates nondimensionalised quantity and the dot differentiation 3 over time. We use ship length L for length scaling, L for mass and L U for time. At first instance, we shall assume that the yaw and sway velocities are restrained from building-up to high values (thus they remain small and the resulting damping forces can be considered as linear) through use of appropriate rudder control. Additionally, ship behaviour is examined at “some distance” from the region of surf-riding, so that, for this first part of the paper, surge velocity is taken as nearly constant. Then, we express the wave terms Ywave  , N (wave) in respect with the frequency of encounter (rather than as functions of absolute wave frequency and position):

Y(wave)  Yw  sin e t 

(3)

N (wave)  N w  cos e t 

(4)

The following notation is applied: Yw , N w are wave force/moment coefficients;  is the ship’s heading relatively to the wave (  0 when the sea is exactly following – generally,  is assumed small). Consider further rudder control with a linear law based on two gains, k1 and k 2 :

k1 multiplies the instantaneous heading deviation from the desired course  r , while k 2′ multiplies yaw’s angular velocity:

  k1 (  r )  k 2 r 

(5)

Substituting (3), (4) and (5) in (1) and (2), uncoupling yaw from sway and using well known expressions for system gain and time constants, K , T1 , T2 , T3, the following differential equation of heading angle is obtained:

  b  p1  f cose t     q 2 1  hcose t      j

(6)

366 K.J. Spyrou

The above third-order differential equation has time-dependent coefficients in two places. As is well known however, if T1 is much greater than T2 and T3 , we can use the so-called simplified yaw response model of Nomoto et al (1957). In that case the order of equation (6) is reduced by one:

T      K   A cos(e t   a)

(7)

K , T  are respectively system gain and time constants,  is relative heading

angle (assumed small),  is rudder angle, A is wave excitation amplitude, e is the encounter frequency that is calculated here with reference to the desired heading and is thus independent of the oscillatory yaw motion; and a is a phase angle. Coupling (7) with the autopilot (5) and drop for simplicity of the phase angle a , leads to:

      0 ( yaw) 2 [1  h cos(e t )]  j In the above formulation,  0 ( yaw ) 

(8)

k 1 K  T  ,   1  k 2 K  T  (damping),

h  A k1 K  (amplitude of parametric variation of restoring), j  k1 K  r T  . It is easily recognized that (8) is Mathieu’s equation with the addition however of biaslike external static forcing term, j . For stability, positive T  is required as 1 T  is the inverse of the damping of the unsteered vessel. However, large positive T  implies slow convergence towards the corresponding steady rate-of-turn which is determined by the value of the static gain K  . A trend exists for large T  to appear in conjunction with large K  which gives a nearly straight-line spiral curve. The effect of active control on damping is represented by the quantity k 2 K  T  . It depends thus on the yaw rate (“differential”) gain term in the autopilot. If T   0 , suitable choice of k 2 can turn the damping of the system positive since k 2 multiplies the positive quantity K  T  , thereby yielding stability for the steered ship in calm sea. The wave effects are lumped into the restoring and independent-periodic-forcing terms since the quantities K  and T  were assumed to be, at first approximation, unaffected by the wave. If the amplitude of wave excitation A exceeds k1 K  then, on the basis of (8), negative yaw restoring will arise around the trough. Should the duration of operation under negative restoring be long enough, undesired turning motion will be initiated (“broaching-to”). From a dynamics perspective there is complete equivalence with a capsize event of the so-called “pure-loss” type. It can be avoided if the proportional gain k1 is chosen to be always greater than A K  even for the most extreme wave environment where the ship will operate (it should be a matter of further investigation to what extend this is technically feasible). A notable difference between the manifestation of this instability in roll

Parallels of Yaw and Roll Dynamics of Ships in Astern Seas 367

and in yaw is that in roll it arises near the crest of the wave, whereas in yaw the ship becomes vulnerable around the trough. We may rewrite (8) on the basis of heading error  1    r and then apply the time transformation s   0 ( yaw) t  :

d 2 1 ds

2

 2

d 1  [1  h cos(s )]  f cos s ds

Where   e  0 ( yaw) ,

h  A / k1 K  and

(9)

f  A  r (T   0 ( yaw) 2 ) . The

damping ratio is given by the expression: 2  1  k 2 K 

k1 K  T  (the presence

of k1 should be noted in  ).

3 Comparison with the Roll Equation It is obvious from (9) that parametric instability of yaw may also arise, very much like that of roll. To establish the analogy we remind that the generic equation of roll in a “following” sea linearised in terms of the roll angle is:

d 2  d

2

 2

d   1  h cos    0 d

(10)

  is the scaled roll angle,      v , with  the true roll angle and  v the angle of vanishing stability. Also,   0( roll) t . The damping ratio is given in this

case by, 2  B 0( roll ) M g (GM )  where B is the dimensional linear damping coefficient, M is ship mass and GM  is the metacentric height. Roll’s natural frequency is 0( roll )  Mg (GM ) ( I  I ) . With substitution of 0( roll ) we

obtain further that 2  B ( I  I ) M g (GM ) . Also, the amplitude of the parametric term is h   GM  GM  where  GM  is the difference of metacentric height at the crest from the still water value. This is a common assumption but of course it results in a highly idealised formulation because the average (GM ) has no reason to be identical with the still water GM  . In addition, the variation from trough to crest may not be harmonic.

368 K.J. Spyrou

3.1 Conditions for “Exact” Resonance Let us neglect for a while the damping terms of (9) and (10) in order to find out how the Froude number of the vertex of the principal resonance varies between roll and yaw. For overtaking waves the frequency of encounter, e , is positive and the condition of exact resonance is written as: e 0  2 n , n  1,2,3,... where 0 can be the frequency of encounter of either yaw or roll. Thus with increasing n the vertices will accumulate near to zero frequency of encounter. For  r  0 we

can write, e  (2  )c  U  . 

Fn 

 2 L 

(11a)

   1  0( yaw )  n   L   

Considering the domain of variation of the natural frequency of yaw,  0 ( yaw )  k 1 K  T  , for conventional ships the ratio K  T  is usually within

the range 0.3  1.4 [see for example Barr et al. (1981)]. As a matter of fact

 0 ( yaw) should lie in the range [0.55 k1 - 1.18 k1 ]. With a proportional gain k1 between 1.0 and 2.0 ,  0 ( yaw) should be between 0.55 and 1.67 . For roll on the other hand, the natural frequency is nondimensionalised on the basis of ship length and acceleration of gravity, 0  0 L g . The different nondimensionalisation results in a different parametric expression of the critical Froude number:

 Fn 

L  2

 0 ( roll )  

n

L

(11b)

3.2 Relative Magnitudes and Effect of Damping Even when a ship is equipped with bilge keels, the damping ratio is usually quite low. Noted that the roll damping ratio will change if the position of the centre of gravity is modified. As far as yaw is concerned, the damping ratio depends strongly on the autopilot’s gains. Common values are in the range 0.8    1.0 Fossen (1994). Here lies therefore another significant difference between the roll and yaw equations: The yaw damping ratio is normally very large. As a matter of fact, to be placed in one of the regions of resonance there should be very considerable loss of yaw restoring at the trough region. Usually this means that very steep waves will be required. Unfortunately, for such highly damped motion

Parallels of Yaw and Roll Dynamics of Ships in Astern Seas 369

it is not easy to derive simple expressions for the critical parametric amplitude. Even the expression of Gunderson, Rigas & Van Vleck (1974), which is applicable also for large damping values, would not work as  approaches 1.0 :

 02  h  1   2 tanh   2  e2   





(12)

3.2.1 Nonlinearity The dynamic behaviour considering the nonlinearities that in reality exist in the restoring (strong) and in damping (mild) is relatively well understood as it has been studied by a number of investigators, for example Blocki (1980); Zavodney et al. (1990); Soliman & Thompson (1992); Kan (1992); Hamamoto et al. (1995). For the yaw equation, if the autopilot is relatively effective, nonlinearity will reside mainly in the damping.

4 Some Other Aspects of Parametric Rolling 4.1 Bi-Chromatic Waves An interesting and rather little known subject is the effect for parametric rolling occurrence of a quasi-periodic variation of (GZ) (instead of the often considered in a deterministic context periodic one). The consideration of a Mathieu-type equation with internal forcing based on a single frequency represents of course a highly idealised scenario. In Fig. 1 are shown the stability transition curves when two independent frequencies are present. This particular graph was drawn with the second frequency being 75% of the first. Also, the amplitude of parametric forcing at the second frequency was 3 times that of the first. It is noticed that secondary “spikes” have grown on each primary resonance region. Further research has shown that their number tends to increase as the second frequency keeps aloof from the first.

370 K.J. Spyrou

Fig. 1 Boundaries of parametric resonance with two incommensurate frequencies. The inserted graph shows a time realisation of the time-dependent restoring.

4.2 Hardening Restoring Another matter considered was the effect of an initially hardening restoring on the stability transition curves. We have assumed a quintic restoring curve which can be parameterised on the basis of a single parameter  . In scaled form the expression of restoring is: R( )       3  (1   )  5 where   is the scaled roll angle with respect to the vanishing angle. Increase of  means basically stronger initial hardening. The parametric variation was applied only on the linear term. In Fig. 2 are shown the transition curves for restoring which is moderately or strongly hardening. Generally, a process of transformation of the boundary from sharp to ‘brittle’ is in place as  is increased. Fractal-like stability boundaries have been presented earlier for a generic cubic restoring and for a more exact restoring curve (Spyrou 2000a).

Parallels of Yaw and Roll Dynamics of Ships in Astern Seas 371

Fig. 2 Capsize boundaries for increasingly hardening restoring. The left graph is with moderately hardening restoring (   2 ) and the right one is with strongly (   17 ).

5 The Fundamental Nonlinear Surge Dynamics It is well-known that the surging behaviour of a ship in following waves of substantial steepness can become strongly nonlinear for a certain range of speeds (normally for Froude numbers higher than 0.3). This nonlinear behaviour is manifested with a gradual transformation of ship response towards an asymmetric pattern of surging even if the considered wave is of a simple harmonic form. Typically, the ship spends more time on the crests and less on the troughs, a tendency which becomes more pronounced as the speed is increased further. Furthermore, unusual types of behaviour with a stationary nature and in competition with the periodic pattern arise, featuring a forced motion at a speed equal to the wave celerity and with the ship “locked” between two consecutive wave crests. These phenomena have already been studied on the basis of numerical models for a following as well as for a quartering sea environment and specific explanations of their dynamics have been produced (Spyrou 1997). A typical phase-plane plot of nonlinear surging is shown in Fig. 3. We have recently endeavoured to developing a purely analytical description for the surging behaviour, even for the strongly nonlinear range (Spyrou 2000b). This would be very useful for design where closed-form expressions are always preferred. It would benefit also advanced investigations on phenomena such as broaching-to and loss of transverse stability. The differential equation of surging motion has a strong nonlinearity in the stiffness term since the wave force is a sinusoidal function of position. There is

372 K.J. Spyrou

also a weak nonlinearity in the damping (=difference between resistance and thrust). A general form for the equation of surge on a sinusoidal wave is:

m  X u  u  [ Ru, c   T u, c, n ] 

f sin kx   0

(13)

The velocity relatively to the wave is x  u  c. With substitution of suitable polynomial expressions for the thrust and the resistance, eq. (1) becomes,

m  X u  x  3r3 c 2  2r2   2 c  r1    1 n x  3r3 c  r2   2  x 2  r3 x 3  f sin kx  



 

(14)



  2 c 2   1 c n   0 n 2  r1c  r2 c 2  r3 c 3     T c ; n 

R c 

The right-hand-side of (14) reveals the polynomial expressions used for T and R.

Surf-riding: This calculation is straightforward: If  1 

T c; n   R c   1 then 1  sin kx  1 . f

Stationary solutions become possible (surf-riding), located at, x

 T c; n   R c    T c; n   R c   2 1 (2  1) 1  arcsin   arcsin   , x  (15) k k f k k f    

Asymmetric surging: Equation (14) can be brought into the following form, by postulating the substitution of the damping terms with an equivalent quadratic on the basis of a leastsquare fit,

(m  X u ) x   c; n x x  f sin(kx)  T (c; n)  R(c)

(16)

Parallels of Yaw and Roll Dynamics of Ships in Astern Seas 373

For an overtaking wave dx/dt < 0 and the above leads to the following expression for the orbits of the phase plane x, x  ,

2qcos kx  2 psin kx  r dx 1  x   c2 qe 2 pkx   dt k p 1 4 p2



where p 



k m  X u 

, q



(17)

T c; n   Rc k . fk and r  m  X u  m  X u 

The constants c1 and c 2 depend on the initial conditions. The term c 2 qe 2 pkx represents the transient part of the solution and it vanishes gradually since x   (the ship is trailing behind the waves). Therefore the expression for the steady periodic motion is, dx 1  x   dt k

2qcos kx  2 psin kx 

1  4 p  2



r p

(18)

With suitable transformations it can be shown that (18) can be solved explicitly for t, t

m  kx    F , m a  2 

(19)

Where F is the elliptic integral of the first kind, F 





0

1 1  m cos 2 

d with

modulus m . In Fig. 4 is shown the relation between time and position which shows clearly the distortion from the linear pattern. With inversion of (19) we obtain further the following expression for x in terms of t (see also Fig. 5),

 a coskx     1  2 sn 2   t, m 

 m  

(20)

When the ship operates away from the strongly nonlinear regime, the right-handside of (20) tends to obtain the cyclic form cos e t  .

374 K.J. Spyrou

Condition for Homoclinic Connection The surface of a cylinder is the “natural” phase plane for the dynamics of our problem. On such a plane one should note the possibility of a homoclinic connection event evoking the disappearance of the periodic motion. In physical space this could correspond to the realization of global surf-riding which is further linked with the occurrence of “broaching-to”. It could happen as soon as the unstable equilibrium near to the wave crest “collides” with the periodic orbit. Unstable stationary point:

x  0 x

2  1 k



1 T c; n   Rc  sin 1 k f

(21)

The steady periodic orbit is given by eq. (18). Substitution of (21) into (18) yields the following expression of critical amplitude for the surge wave force,

f crit 

Rc; n   T n  2 k m  X u 2





2

 4 2

(22)

. x

x

Fig. 3 Typical phase-portrait of surf-riding

Parallels of Yaw and Roll Dynamics of Ships in Astern Seas 375 2

4

6

8

10

kx

-2 -4 -6 -8

t Fig. 4 During asymmetric surging the relation between b time and position is distorted from the straight line. 1

cos(kx)

0.5 t 2

4

6

8

10

12

14

-0.5

-1 Fig. 5 Nonlinear surging according to eq. (20).

6 The Effect of Nonlinear Surging on Roll As is well known, an implicit assumption made in the analysis of pure-loss of stability and parametric rolling is that the forward speed is constant. Such an assumption is not always valid. In principle, for dangerous roll behavior, steep and long waves are required. Waves of this kind will incur however also significant nonlinear effects on surge. The characteristic of large-amplitude surging is that it is asymmetric and the ship stays longer near the crests than near the troughs. This effect is imported into the yaw and roll dynamics through the restoring terms of the corresponding equations. The nonlinearity of surge is detrimental for roll stability because around the crest (where the ship stays longer) restoring capability is reduced. For yaw on the other hand, the effect is opposite. Yaw stability is not

376 K.J. Spyrou

worsened because the passage of the ship from the trough is quicker. The danger arises in steeper waves and especially during the process of capture in surf-riding. The three main forces acting in the surge direction are the resistance, the wave and the propulsion force. As is well known these result in a pendulum-like equation for surge the exact form of which may be found for example in Spyrou (2000b). This surge equation should be solved simultaneously with the following equation of roll (or that of yaw for broaching-to):

  2     02 1  h coskx    q 3  0

(23)

Note that the cosine is written in terms of the position on the wave, x , rather than in terms of time. The coupling to the surge equation exists because of the presence of x in the restoring term of the above roll equation. We can show now how the transition curves are modified when roll is coupled with surge.

Fig. 6 Coupled surging and rolling leading to capsize

The calculations were based on a ship with  0( roll )  0.84 (  0 ( roll )  1.577 ) and

  0.0585 . We have examined whether the normalised roll angle   exceeds the value of 1 (from an initial perturbation 0.01 and with zero initial velocity) within a specific amount of time ( t  200 s). In Fig. 6 is presented an example of time realisation for the coupled surging and rolling leading to capsize when strong nonlinear effects in surge are present. Surge motion can have a very profound effect on the “capsize” domains as realised from the stability diagram of Fig. 7. Rather than plotting the frequency ratio versus the parametric amplitude, we found more relevant to present the information in terms of the Froude number. For the considered ship, the principal resonance cannot be realised in following waves because the required Froude number is negative (the ship should be backing rather than going forward). The lower part of the fundamental is the only place where there is some commonality with the conventional (‘damped’) Ince-Strutt diagram. The upper part of the fundamental tends to become considerably wider. The

Parallels of Yaw and Roll Dynamics of Ships in Astern Seas 377

immediately next resonance occupies an enlarged domain; but the following two seem to degenerate. This may relate with the emergence of a surf-riding domain where the behaviour of the ship is stationary and travels with the wave with its centre located near a trough. 1.5 h 1.4 1.3 1.2 1.1 1 0.9

0.1

0.15

0.2

0.25 0.3 0.35 Nominal Fn

Fig. 7 Capsize boundary with nonlinear surging

7 Concluding Remarks When roll stability in a following sea is examined, it is common to distinguish between two mechanisms of capsize: pure-loss of stability, where the ship departs from the state of upright equilibrium due to negative restoring on a wave crest. Then heel increases monotonically until the ship is overturned. In this mode the magnitude of damping plays little role. In parametric instability on the other hand, which is near to the classical Mathieu-type mechanism, the build-up is oscillatory and the magnitude of damping is very important. Instabilities of a similar nature are possible in yaw, resulting in the behaviour known as broaching-to: The parallel to the pure-loss mechanism can be termed as broaching-to due to surfriding. It can happen at Froude numbers relatively near to the wave celerity. The parametric-type mechanism of broaching-to is relevant for lower Froude numbers but requires higher wave steepness. Extra forcing terms and couplings sometimes alter significantly the domains of instability. Surging dynamics can affect both mechanisms of roll instability and this effect deserves further investigation.

378 K.J. Spyrou

References Barr R, Miller ER, Ankudinov V and Lee FC (1981) Technical basis for manoeuvring performance standards. Technical Report 8103-3, Hydronautics, Inc, submitted by the USA to the Int Marit Organ (IMO). Blocki W (1980) Ship safety in connection with parametric resonance of the roll, Int Shipbuilding Prog, 27:36-53. Clarke D, Gendling P and Hine G (1983) Application of manoeuvring criteria in hull design using linear theory, Trans. RINA, 125:45-68. Fossen TI (1994) Guidance and Control of Ocean Vehicles, John Wiley and Sons, Chichester, UK. Grim O (1952) Rollschwingungen, Stabilität und Sicherheit im Seegang, Schiffstechnik, 1, 1:10-21. Gunderson H, Rigas H and Van Vleck FS (1974) A technique for determining stability regions for the damped Mathieu equation. SIAM J of Appl Math, 26, 2:345-349. Hamamoto M, Umeda N, Matsuda A and Sera W (1995) Analysis of low-cycle-resonance in astern seas. J of the Soc of Naval Archit of Japan, 177:197-206. Kan M (1990) Surging of large amplitude and surf-riding of ships in following seas. Selected Papers in Naval Archit and Ocean Eng, The Soc of Naval Archit of Japan. Kan M (1992) Chaotic capsizing. Proceedings Osaka Meeting on Seakeeping Performance, 20 th ITTC Seakeeping Committee and Kansai Fluid-Dynamics Res Group, 155-180. Kerwin JE (1955) Notes on rolling in longitudinal waves, Int Shipbuilding Prog, 2, 16:597-614. Nomoto K, Taguchi K, Honda K and Hirano S (1957) On the steering qualities of ships. Int Shipbuilding Prog, 4, 35, July, 354-370. Paulling JR (1961) The transverse stability of a ship in a longitudinal seaway, J of Ship Res 4:37-49. Soliman MS and Thompson JMT (1992) Indeterminate sub-critical bifurcations in parametric resonance, Proc of the Royal Soc of London, A, 438, 511-518. Spyrou KJ (1996) Dynamic instability in quartering waves: the behaviour of a ship during broaching. J of Ship Res, 40, 1, 46-59. Spyrou KJ (1997) On the nonlinear dynamics of broaching-to. Proc, Int Conf on Design for Abnormal Waves, The Royal Instof Naval Archit, Glasgow, October, 12 pages. Spyrou KJ (2000a) Designing against parametric instability in following seas, Ocean Eng, 27, 6, 625-654. Spyrou KJ (2000b) On the parametric rolling of ships in a following sea under simultaneous nonlinear periodic surging. Philos Transactions of the Royal Soc of London, Α 358:1813-1834. Zavodney LD, Nayfeh AH and Sanchez (1990) Bifurcations and chaos in parametrically excited single-degree-of-freedom systems. Nonlinear Dyn 1, 1:1-21.

Model Experiment on Heel-Induced Hydrodynamic Forces in Waves for Realising Quantitative Prediction of Broaching Hirotada Hashimoto*, Naoya Umeda* and Akihiko Matsuda** *Osaka University, Yamadaoka, Suita, Osaka, 565-0871, Japan ** National Research Institute of Fisheries Engineering, Hasaki, Kamisu, Ibaraki, 314-0408, Japan

Abstract Mathematical models for capsizing due to broaching, one of the great threat to ships running in following and quartering seas, have been developed by many researchers. However these models can predict it only qualitatively. For realising quantitative prediction of broaching, we have conducted systematic captive model experiments with large heel angles up to 50 degrees in severe following waves. As a result, the details of nonlinear heel-induced hydrodynamic forces with respect to heel angle in waves are presented. Then comparisons between a free running model experiment and the numerical simulation with direct use of the measured heel-induced hydrodynamic forces are carried out. As a result, the mathematical model for realising quantitative prediction of broaching in severe following and quartering seas is presented. Finally the simplified mathematical model is also proposed for more practical uses with less experimental efforts.

Nomeclature c Fn GZ H Ixx Izz Jxx Jzz KMNL KRNL KRWNL Kp Kr KrW KR Kv

wave celerity nominal Froude number righting arm wave height moment of inertia in roll moment of inertia in yaw added moment of inertia in roll added moment of inertia in yaw nonlinear manoeuvring forces in roll nonlinear heel-induced hydrodynamic roll moment in still water nonlinear heel-induced hydrodynamic roll moment in waves derivative of roll moment with respect to roll rate derivative of roll moment with respect to yaw rate wave effect on the derivative of roll moment with respect to yaw rate rudder gain derivative of roll moment with respect to sway velocity

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_21, © Springer Science+Business Media B.V. 2011

379

380 H. Hashimoto et al. KvW Kw K KW K’ L m mx my n NMNL NRNL NRWNL Nr NrW Nv NvW Nw N NW N’ p r R t T TD TE TW u v XMNL XRNL X’w Xrud X’ YMNL YRNL YRWNL Yr YrW Yv YvW Yw Y YW Y’ zH

 c   

wave effect on the derivative of roll moment with respect to sway velocity wave-induced roll moment derivative of roll moment with respect to rudder angle wave effect on the derivative of roll moment with respect to rudder angle K’=K/(1/2Ld2u2) ship length between perpendiculars ship mass added mass in surge added mass in sway propeller revolution number nonlinear manoeuvring forces in yaw nonlinear heel-induced hydrodynamic yaw moment in still water nonlinear heel-induced hydrodynamic yaw moment in waves derivative of yaw moment with respect to yaw rate wave effect on the derivative of yaw moment with respect to yaw rate derivative of yaw moment with respect to sway velocity wave effect on the derivative of yaw moment with respect to sway velocity wave-induced yaw moment derivative of yaw moment with respect to rudder angle wave effect on the derivative of yaw moment with respect to rudder angle N’=N/(1/2L2du2) roll rate yaw rate ship resistance time propeller thrust time constant for differential control time constant for steering gear wave effect on propeller thrust surge velocity sway velocity nonlinear manoeuvring forces in surge nonlinear heel-induced hydrodynamic surge force in still water wave-induced surge force with empirical correction rudder-induced surge force X’=X/(1/2Ldu2) nonlinear manoeuvring forces in sway nonlinear heel-induced hydrodynamic sway force in still water nonlinear heel-induced hydrodynamic sway force in waves derivative of sway force with respect to yaw rate wave effect on the derivative of sway force with respect to yaw rate derivative of sway force with respect to sway velocity wave effect on the derivative of sway force with respect to sway velocity wave-induced sway force derivative of sway force with respect to rudder angle wave effect on the derivative of sway force with respect to rudder angle Y’=Y/(1/2Ldu2) vertical position of centre of sway force due to lateral motions heading angle from wave direction desired heading angle for auto pilot rudder angle roll angle wave length

Model Experiment on Heel-Induced Hydrodynamic Forces 381   G G

pitch angle water density longitudinal position of centre of gravity from a wave trough vertical distance between centre of gravity and still water plane

1 Introduction Recent model experiments (e.g. Umeda et al. 1999) demonstrate that a ship complying with the current Intact Stability Code (IS Code) of the International Maritime Organisation (IMO) rarely capsizes in non-breaking beam waves but could occasionally capsize when she runs in following and quartering seas. Although the IMO circulated a simple guidance applicable to all ships for avoiding danger in following and quartering seas, real capsizing boundaries might depend on detailed particulars of each ship. By responding to these situations the Sub-Committee on stability, load lines and on fishing vessel safety has started to review the IS Code. On this revision, an alternative approval of safety with direct assessment by physical or numerical tests is considered. This performance-based assessment of safety is especially expected to prevent ship capsizing due to broaching in following and quartering seas. This is because the existing stability criteria cannot deal with this phenomenon which depends on hull form details as well as operational practices. At this stage numerical models are required to provide not only qualitative agreement but also quantitative one with help of model experiments. Toward this direction, the International Towing Tank Conference (ITTC) had conducted a benchmark test of several numerical models by comparing them with existing capsizing model experiments in following and quartering seas, which cover ship capsizing due to parametric rolling and broaching. As a result, it was confirmed that only a few numerical models could qualitatively predict capsizing due to broaching and none could do it quantitatively (Umeda et al. 2001). Therefore, existing numerical modelling techniques should be upgraded to realise the quantitative prediction of broaching. For this purpose, it is necessary to systematically examine all factors relevant to capsize due to broaching in following and quartering seas further. It has been pointed out that the effect of heel-induced manoeuvring forces in calm water is significant for broaching prediction (Renilson et al. 2000). This effect was taken into account as a linear function of roll angle and their derivatives were obtained from a captive model test for small heel angle in calm water. However the relationship between heel-induced hydrodynamic forces and a roll angle could be nonlinear because under-water hull form becomes quite asymmetric with increasing of roll angle. Therefore it is important to conduct a model experiment to measure heel-induced hydrodynamic forces especially for large heel angle. However this kind of experiment is quite difficult because of the limitation of the experimental setup. Within a framework of ordinary experimental procedure and setup, problems of waterinflow and water-accumulation in a hull cannot be avoided. To overcome these problems,

382 H. Hashimoto et al.

using a complete watertight ship model or devices to prevent a dynamometer to be flooded is required. In past Nakato et al. conducted a captive model experiment with special setup that a dynamometer is settled quite higher position than the centre of ship gravity to measure restoring moments in waves with a wide range of roll angle. (Nakato et al. 1982) However their experimental method and results were not available to practical use because a dynamometer settled at high position provides significant trim moment during towing. For these reasons, a captive model experiment to measure hydrodynamic forces induced by large heel is quite difficult even in calm water. Therefore it is important to develop an experimental procedure to directly measure these forces in both calm water and waves. In our previous work, we had proposed a new experimental method with a purpose-built ship model and an experimental setup, which can realise captive tests with up to 90 degrees of heel. (Hashimoto et al. 2004) Then captive tests were conducted to measure heel-induced hydrodynamic forces in calm water to discuss its importance on broaching prediction. In this paper captive model tests to measure heel-induced hydrodynamic forces in heavy following waves were conducted. Based on these results, the mathematical model was upgraded by introducing a nonlinear model of the hydrodynamic forces as a function of roll angle, and was applied to the prediction of ship motions in following and quartering waves. Then comparisons of the 4 DOF mathematical models with and without this effect were conducted, together with the existing free-running experiments, to examine the importance of nonlinear heelinduced hydrodynamic forces in waves on the prediction of ship capsizing due to broaching in following and quartering seas.

2 Captive Model Experiments with Large Heel 2.1 Procedure of a Model Experiment For measuring heel-induced hydrodynamic forces with large heel angles, a new 1/25 scaled model of the 135GT Japanese purse seiner known as the ITTC Ship A-2 was built. The Body plan and general arrangement of the ship model and the principal particulars are shown in Fig. 1, Fig. 2 and Table. 1, respectively. Here small model size was selected by considering a maximum loading allowance of the dynamometer in case whole of a ship model is in the air. The ship model is completely watertight and has bulwarks, freeing ports and the super structures that are similar to the ship model used in the free-running model experiment (Umeda et al. 1999). There are two poles, having a scale, that fix the model and adjust a model attitude. Each pole has gimbals at their bottom to realize a specified heel angle. Captive model experiments were conducted at the seakeeping and manoeuvring basin of National Research Institute of Fisheries Engineering (NRIFE). The model was towed by a main towing carriage in long-crested regular waves, and model was equipped with

Model Experiment on Heel-Induced Hydrodynamic Forces 383

a rudder but without a propeller. The model was completely fixed by two poles in all directions. Since a model attitude is fixed, only one data for certain relative position of a wave can be obtained per one run of towing carriage. In this experiment heel-induced hydrodynamic forces in waves are identified by measuring eight relative positions of a wave with the same heel angle. The surge force, sway force, heave force, roll moment, pitch moment and yaw moment acting on the towed model were detected by a dynamometer located in vertically high position where no watersplash happens.

Fig. 1 Body plan of the subject ship

Fig. 2 General arrangement of the purpose-built ship model Table 1 Principal particulars of the subject ship Item

Values

length : Lpp

34.5 m

breadth : B

7.60 m

depth : D

3.07 m

mean draught : d

2.65 m

block coefficient : Cb

0.597

longitudinal position of centre of

1.31 m

gravity from the midship : xCG

aft

metacentric height : GM

1.00 m

natural roll period : T

7.4 s

rudder area : AR

3.49 m2

time constant of steering gear : TE

0.63 s

proportional gain: KP

1.0

time constant for differential control: TD

0.0 s

maximum rudder angle: max

 35 deg

384 H. Hashimoto et al.

The procedure of the experiment is as follows. Firstly instantaneous heave and pitch with no heel are estimated for certain relative position of a wave from the model tests with conventional setup in moderate wave steepness. Since wave steepness is different from a desired value, we estimate a ship instantaneous heave and pitch by assuming linearity of them with respect to wave steepness. The effect of heel is added with the Froude-Krylov assumption. Here the calculation is obtained by integrating the undisturbed wave pressure up to the wave surface with the Smith effect for free heave and pitch. By adjusting length of two poles and angles of gimbals, the sinkage, trim and heel angle are set to be equal to the estimated values. An example photograph of the experimental setup is shown in Fig. 3. Here the estimated ship attitude does not usually coincide with the real one. However it is quite difficult to accurately estimate a ship attitude with large heel at high Froude number in steep waves. To correct the difference of a running attitude, additional experiments are required for obtaining the derivatives of heel-induced hydrodynamic forces with respect to heave and pitch. If we conduct this additional experiment to all relative positions and heel angles, the number of model runs could be prohibitively large. To avoid this problem, we apply the following method. As mentioned we repeat eight model runs to measure heel-induced hydrodynamic forces for one wave cycle. Since a ship model is slowly overtaken by waves, the other seven data measured with undesired ship attitude is available to obtain the derivatives of hydrodynamic forces with respect to heave and pitch at specified relative position. Using such derivatives we can reasonably estimate all forces and moments under the accurate attitude in heave and pitch by solving the simultaneous equations satisfying the heave force and pitch moment equal to zero.

Fig. 3 Setup of the captive model experiment

2.2 Procedure of an Analysis The surge force, X, sway force, Y, heave force, Z, roll moment, K, pitch moment, M and yaw moment, N, were measured as functions of the heave, pitch, roll, Froude number and horizontal position of ship to a wave trough, G/. Wave height at the ship centre of gravity was also measured by a servo-needle wave probe. The

Model Experiment on Heel-Induced Hydrodynamic Forces 385

definition of directions of the measured forces and moments is shown in Fig. 4. All forces and moments were measured in horizontal axes. Each moment is geometrically converted from the measured value around the centre of the dynamometer to that of the ship gravity. By assuming the additional change in heave and pitch are small the measured forces and moments can be expanded as Eqs. (1)-(6).   N

X

Y

K

Z

M

N Z

Fig. 4 Definition of directions of measured forces and moments

X ( G , ; , Fn , G /  )  X ( G 0 ,0 ;  , Fn , G /  )  X  ( G 0 ,0 ;  , Fn , G /  )   G *  X  ( G 0 ,0 ;  , Fn , G /  )   *

Y ( G , ; , Fn , G /  )  Y ( G 0 ,0 ; , Fn , G /  )  Y  ( G 0 ,0 ; , Fn , G /  )   G *  Y  ( G 0 ,0 ; , Fn , G /  )  * Z ( G , ; , Fn , G /  )  Z ( G 0 ,0 ;  , Fn , G /  )  Z  ( G 0 ,0 ;  , Fn , G /  )   G *  Z  ( G 0 ,0 ; , Fn , G /  )   * K ( G , ;  , Fn , G /  )  K ( G 0 ,0 ;  , Fn , G /  )  K  ( G 0 ,0 ;  , Fn , G /  )   G *  K  ( G 0 , 0 ;  , Fn , G /  )   * M ( G , ;  , Fn , G /  )  M ( G 0 ,0 ; , Fn , G /  )  M  ( G 0 , 0 ; , Fn , G /  )   G *  M  ( G 0 ,0 ; , Fn , G /  )   * N ( G , ; , Fn , G /  )  N ( G 0 ,0 ;  , Fn , G /  )  N  ( G 0 ,0 ;  , Fn , G /  )   G *  N  ( G 0 ,0 ; , Fn , G /  )   *

(1)

(2)

(3)

(4)

(5)

(6)

Here  G*   G   G 0

(7)

 *   0

(8)

where G0 and 0 ni dicate the heave and pitch angle, respectively, initially estimated. If a ship model is free in heave and pitch, the heave force and pitch moment should be zero. Therefore G*,  * can be obtained by solving the following simultaneous equations.

386 H. Hashimoto et al.

Z ( G 0 ,0 ;  , Fn , G /  )  Z  ( G 0 ,0 ;  , Fn , G /  ) G *  Z  ( G 0 ,0 ;  , Fn , G /  ) *  0

(9)

M ( G 0 ,0 ; , Fn , G /  )  M  ( G 0 ,0 ; , Fn , G /  ) G *  M  ( G 0 ,0 ; , Fn , G /  ) *  0 (10)

Then, the surge force and sway force, roll moment and yaw moment for the case in which model is free in heave and pitch can be estimated with the Equations (1), (2), (4) and (6).

3 Experimental Results Captive model experiments to measure heel-induced hydrodynamic forces in severe following waves were conducted. The wave condition was selected to correspond to that of the ITTC Benchmark Testing Programme, H/=1/10 and /L=1.637. The captive tests were conducted for various sets of heel angle at Froude number of 0.4. The here heel angles were 0, 10, 20, 30, 40 and 50 degrees. The heel-induced hydrodynamic surge force, sway force, yaw moment and roll moment in waves can be obtained by subtracting a constant value obtained in calm water with each heel angle. Finally these results are identified by the Fourier expansion with encounter frequency. An example photograph of the experiment is shown in Fig. 5.

Fig. 5 A photograph of the captive test in waves

The experimental results of non-dimensional heel-induced surge force, X’, sway force, Y’, yaw moment, N’, and roll moment, K’, in following waves at Fn=0.4 are shown in Figs. 6-9. The amplitude of X’ linearly increases with respect to roll angle and the phase of X’ drastically changes. Since the transverse hull asymmetry has a secondary effect on a longitudinal force, the amplitude of X’ itself is much smaller than the amplitude of wave-induced surge force for upright condition. Therefore the heel-induced surge force in waves can be neglected for numerical prediction. The amplitude of Y’ linearly increases up to 20 degrees but does nonlinearly over 20 degrees. The phase of Y’ also changes with respect to roll angle. The N’ and K’ have similar tendency in its change as the Y’. Since the

Model Experiment on Heel-Induced Hydrodynamic Forces 387

K’ could be important for capsizing prediction, a comparison of roll restoring moment in waves among the experimental result, the Froude-Krylov calculation on its own and the sum of the Froude-Krylov calculation and the experimental formula of lift effect (Umeda et al. 2002). Here the Froude-Krylov calculation is done by integrating the undisturbed wave pressure up to the wave surface with the Smith effect for free heave and pitch. Fig. 10 shows a comparison of GZ curve between the experiment and the two calculations. The Froude-Krylov calculation overestimates the amplitude of GZ variation. There is significant difference in the average value and the phase of GZ variation. The experimentally obtained relative position of the minimum GZ is not a wave crest but a wave down-slope. The calculation by the Froude-Krylov assumption corrected by the experimental formula can predict these tendencies. However the difference from the experimental result is not negligibly small from a viewpoint of realising the quantitative prediction. Therefore a proposed captive model experiment could be recommended to accurately predict the wave effect on roll restoring moment for large heel angle in severe waves. 0.08 0.06 0.04 o

 = 10

0.02  X'

o

 = 20

0.00

o

 = 30

o

 = 40

-0.02

o

 = 50

-0.04 -0.06 -0.08 0.00

0.25

0.50

0.75

1.00

 G /

Fig. 6 Experimental result of non-dimensional heel-induced surge force with H/=1/10, /L=1.637, =0 degrees and Fn=0.4 0.3 0.2 0.1

o

 = 10

o

 Y'

 = 20

0.0

o

 = 30

o

 = 40

o

-0.1

 = 50

-0.2 -0.3 0.00

0.25

0.50

0.75

1.00

 G /

Fig. 7 Experimental result of non-dimensional heel-induced sway force with H/=1/10, /L=1.637, =0 degrees and Fn=0.4

388 H. Hashimoto et al. 0.100 0.075 0.050 o

 = 10

0.025  N'

o

 = 20

0.000

o

 = 30

o

 = 40

-0.025

o

 = 50

-0.050 -0.075 -0.100 0.00

0.25

0.50

0.75

1.00

 G /

Fig. 8 Experimental result of non-dimensional heel-induced yaw moment with H/=1/10, /L=1.637, =0 degrees and Fn=0.4 0.4 0.3 0.2 o

 = 10

0.1  K'

o

 = 20

0.0

o

 = 30

o

 = 40

-0.1

o

 = 50

-0.2 -0.3 -0.4 0.00

0.25

0.50

0.75

1.00

 G /

Fig. 9 Experimental result of non-dimensional heel-induced roll moment with H/=1/10, /L=1.637, =0 degrees and Fn=0.4 0.4 0.3

GZ (m)

0.2 0.1 0.0 0.00

0.25

0.50

0.75

-0.1 -0.2 -0.3

1.00 G/ 

EXP. F.K. F.K. with lift

Fig. 10 Comparison of righting arm in waves between the experiment, the Froude-Krylov calculation and the sum of Froude-Krylov calculation and the experimental formula of lift effect with H/=1/10, /L=1.637, =0 degrees, Fn=0.4 and =20 degrees

Model Experiment on Heel-Induced Hydrodynamic Forces 389

4 Mathematical Modelling The mathematical model of surge-sway-yaw-roll motions was developed by Umeda and Renilson (Umeda et al. 1992) and Umeda (Umeda 1999) for capsizing associated with surf-riding in following and quartering waves. The details of this model can be found in the literature (Umeda et al. 2002). Since this model had consistently ignored higher order terms of waves, several higher order terms were added for improving the prediction accuracy (Hashimoto et al. 2004). Then the heel-induced hydrodynamic forces measured in waves are newly added in this research. Two coordinate systems used here are shown in Fig. 11: (1) a wave fixed with its origin at a wave trough, the  axis in the direction of wave travel; and (2) an upright body fixed with its origin at the centre of ship gravity. 

x Ship



G O



y



G

wave trough



O

y

G



Z 

Fig. 11 Coordinate systems

The state vector, x and control vector, b , of this system are defined as follows. x  ( x1 , x 2 ,  , x8 ) T  { G /  , u , v,  , r ,  , p,  }T

(11)

b  {n,  c }T

(12)

The dynamical system can be represented by the following state equation. x  F(x;b)  { f1 (x;b), f 2 (x;b),, f 8 (x;b)}T

(13)

where f1 (x;b)  (u cos   v sin   c) / 

(14)

f2 (x;b)  {T (u ; n )  T W (G / , u ,  ; n )  R (u )  X R N L (u ,  )  X M N L (u , v , r )  X rud (G /  , u ,  ,  ; n )  X w (G / ,  )}/ (m  m x )

(15)

390 H. Hashimoto et al. f3 (x;b)  {(m  m x )u r  Y v (u ; n )v  Y v W (G /  , u ,  )v  Y r (u ; n )r  Y r W (G /  , u ,  )r Y

M NL

(u , v , r )  Y

R NL

(u ,  )  Y

RW NL

(G /  ,  ,  )  Y  (u ; n )  Y  W (G /  , u ,  ; n )

(16)

Y w (G /  , u ,  ; n )} / (m  m y )

f 4 ( x; b )  r f5 (x;b)  {N v (u ; n )v  N v W (G /  , u ,  )v  N r (u ; n )r  N r W (G /  , u ,  )r N

M NL

(u , v , r )  N R N L (u ,  )  N R W N L (G /  ,  ,  )  N  (u ; n )  N  W (G /  , u ,  ; n )

(17) (18)

 N w (G /  , u ,  ; n )} / ( I ZZ  J Z Z ) f 6 ( x; b )  p

f7 ( x;b)  [m x z H (u , v )u r  K v (u , v ; n )v  K v W (G /  , u , v ,  )v  K r (u , v ,; n )r  K r W (G /  , u , v ,  )r  K M N L (u , v , r )  K P (u ) p  K R N L (u ,  )  K R W N L (G /  ,  ,  )

(19)

 K  (u ; n )  K  W (G /  , u ,  ; n )  K w (G /  , u ,  ; n )  m gGZ ( )] / ( I xx  J xx )

f 8 (x;b)  {  K R (    C )  K R TD r} / TE

(20)

The underlined parts indicate the wave effect on heel-induced hydrodynamic forces newly added. Here the Grim’s effective wave concept (Grim 1961) is used for the estimation of heel-induced hydrodynamic forces in waves with heading angle. In a numerical simulation, newly measured heel-induced hydrodynamic forces in calm water are also taken into account. Although the experimental results shown in Figs. 6-9 are obtained for only one Froude number, we assume that these results are applicable to all Froude numbers because of the limitation of available experimental data. In the previous work, the wave effect on roll restoring moment is considered as the sum of the Froude-Krylov calculation and the experimental formula of lift effect (Umeda et al. 2002). However the estimation accuracy is not satisfactory as shown in Fig.10. Therefore the measured heel-induced hydrodynamic roll moment in waves is directly used in the mathematical model. Throughout this paper, we call this model including all higher order terms “enhanced model”.

5 Numerical Results Firstly comparisons between the numerical results with nonlinear heel-induced hydrodynamic forces in waves and without them (Hashimoto et al., 2004) as well as the existing free running model experiments (Umeda et al., 1999) were conducted. The comparison in the case that the ship experiences a periodic motion is shown in Fig.12. The difference between the calculated results with and without the wave effect

Model Experiment on Heel-Induced Hydrodynamic Forces 391

on heel-induced hydrodynamic forces is not so significant because its maximum roll angle is not so large. The comparison in the case that the ship suffers surf-riding, broaching and capsizing is shown in Fig. 13. The mathematical model with the nonlinear heel-induced hydrodynamic forces in waves provides better agreement with the model experiment to some extent. The comparison in boundaries of ship motion modes for control parameters of auto pilot course and nominal Froude number is shown in Fig.14. The procedure of this calculation can be found in the literature (Umeda et al. 2002). In the numerical results, a region of capsizing due to broaching becomes smaller and that of stable surf-riding does larger like the experimental result. These comparisons demonstrate that the prediction accuracy becomes better by taking the heel-induced hydrodynamic forces in waves into account, and the quantitative prediction of broaching and capsizing in heavy following and quartering waves is realised by the enhanced model. Secondly a comparison between the enhanced model and that without heelinduced hydrodynamic sway force and yaw moment in waves was conducted to examine the importance of roll restoring variation for large heel angle on capsizing prediction. The comparison in boundaries of ship motion modes for control parameters is shown in Fig. 15. In the numerical result, a region of capsizing due to broaching becomes larger and that of stable surf-riding does smaller when the wave effect on heel-induced hydrodynamic sway force and yaw moment are neglected. However the difference between two calculations is not so significant as compared to Fig. 14. This result indicates that the wave effect on heel-induced roll moment at large heel angle is important for accurate prediction of capsizing boundary. 20

Pitch(degrees)

20

0 0 -10

5

10

15

20

t(s)

-20

5

10

15

20

t(s)

-40 -60

Yaw(degrees)

60 40 20 0 -200

5

10

15

20

t(s)

Rudder(degrees)

5

5

10

15

20

t(s)

-20 60 40 20 0 -200

5

10

15

20

t(s)

-40 -60

Yaw(degrees)

-40 -60

45 30 15 0 -150

0 0 -10

Roll(degrees)

Roll(degrees)

60 40 20 0 -200

-30 -45

Pitch(degrees)

10

10

60 40 20 0 -200

5

10

15

20

t(s)

-40 -60

Rudder(degrees)

10

15

20

t(s)

45 30 15 0 -150 -30 -45

5

10

15

20

t(s)

Experiment Calculation

Fig. 12 Comparison of numerical results with H/=1/10, /L=1.637, Fn=0.3, c=-30 degrees. (left: without heel-induced hydrodynamic forces in waves, right: with them)

392 H. Hashimoto et al. Pitch(degrees)

20 10 0 0 -10 -20 90 60 30 0 -300 -60 -90

Pitch(degrees)

5

10

15

20

t(s) Roll(degrees)

10

15

20

t(s)

Yaw(degrees)

10

15

20

t(s)

90 60 30 0 -300 -60 -90

5

10

15

20

t(s)

Yaw(degrees)

5

10

15

20

t(s)

Rudder(degrees)

45 30 15 0 -150 -30 -45

5

Roll(degrees)

5

60 40 20 0 -200 -40 -60

20 10 0 0 -10 -20

5

60 40 20 0 -200 -40 -60

5

10

15

20

t(s)

Rudder(degrees)

10

15

20

t(s)

45 30 15 0 -150 -30 -45

5

10

15

20

t(s)

Experiment Calculation

Fig. 13 Comparison of numerical results with H/=1/10, /L=1.637, Fn=0.43, c=-10 degrees. (left: without heel-induced hydrodynamic forces in waves, right: with them) 0.5

nominal Froude number

nominal Froude number

0.5 0.4 0.3 0.2 0.1 0

10

20

30

40

0.4 0.3 0.2 0.1

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

stable surf-riding periodic motion not identified

0

10

20

30

auto pilot course (degrees)

40

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

Fig. 14 Comparison of boundaries of ship motion modes with H/=1/10, /L=1.637 and the initial periodic state for Fn=0.1 and c=0 degrees. (left: without heel-induced hydrodynamic forces in waves, right: with them)

Model Experiment on Heel-Induced Hydrodynamic Forces 393 0.5

nominal Froude number

nominal Froude number

0.5 0.4 0.3 0.2 0.1 0

10

20

30

40

0.4 0.3 0.2 0.1

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

0

10

20

30

auto pilot course (degrees)

stable surf-riding periodic motion not identified

40

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

Fig. 15 Comparison of boundaries of ship motion modes with H/=1/10, /L=1.637 and the initial periodic state for Fn=0.1 and c=0 degrees. (left: with all higher order terms, right: without heel-induced hydrodynamic sway force and yaw moment in waves)

6 Simplification of Mathematical Model To improve the prediction accuracy of broaching, the authors had investigated higher order terms and executed numerical calculations with and without these terms one by one. Numerical prediction results on the higher order terms were compared with the captive model tests and, when a prediction method for a certain term is not available, the captive test data were directly used for the comparative studies. As a result, quantitative prediction of broaching associated with surf-riding in severe following and quartering waves can be realized by the enhanced model which takes account of higher order terms listed below. a) b) c) d) e) f) g) h) i)

nonlinear manoeuvring forces in calm water (Umeda & Hashomoto, 2002) wave effect on linear manoeuvring forces (Umeda et al., 2003) wave effect on roll restoring moment (Umeda et al., 2003) wave effect on rudder force (Umeda et al., 2003) nonlinear wave forces including mean values (Hashimoto et al., 2004) nonlinear sway-yaw coupling (Hashimoto et al., 2004) wave effect on propeller thrust (Hashimoto et al., 2004) heel-induced hydrodynamic forces for large heel angle in calm water (Hashimoto et al., 2004) wave effect on heel-induced hydrodynamic forces for large heel angle (in this research)

394 H. Hashimoto et al.

By carrying out systematic comparisons of numerical simulations, it was confirmed the following terms are essential among them to realise accurate prediction of dynamic behaviours related to surf-riding and broaching. i) ii) iii) iv) v)

nonlinear manoeuvring forces in calm water. wave effect on linear manoeuvring forces nonlinear wave-induced surge force nonlinear sway-yaw coupling heel-induced hydrodynamic forces for large heel angle in calm water

For the terms i) and iv), conventional captive model tests in manoeuvrability are sufficient. The term ii) can be estimated with a potential theory. For the terms iii) and v), captive mode tests are required but a seakeeping and manoeuvring basin is not necessary. Thus this theoretical prediction method with the minimum captive test data is still practical. Although the heel-induced hydrodynamic forces for large heel angle in waves are also important for quantitative prediction particularly for the capsizing boundary, the captive model test for measuring them is complicated to execute from a practical point of view. Based on the above, the simplified mathematical model for predicting broaching in following and quartering waves are proposed for a practical purpose, which is called “proposed model”. The state equations replaced with the Eqs.(14)-(20) are expressed as follows: f1 (x;b)  (u cos   v sin   c) / 

(21)

f2 (x;b)  {T (u ; n )  T W (G / , u ,  ; n )  R (u )  X R N L (u ,  )  X M N L (u , v , r )

(22)

 X w (G / ,  )}/ (m  m x ) f3 (x;b)  {(m  m x )u r  Y v (u ; n )v  Y v W (G /  , u ,  )v  Y r (u ; n )r  Y r W (G /  , u ,  )r (23) Y M N L (u , v , r )  Y R N L (u ,  )  Y  (u ; n )  Y w (G /  , u ,  ; n )} / (m  m y )

f 4 ( x; b )  r

(24)

f5 (x;b)  {N v (u ; n )v  N v W (G /  , u ,  )v  N r (u ; n )r  N r W (G /  , u ,  )r

(25)

 N M N L (u , v , r )  N R N L (u ,  )  N  (u ; n )  N w (G /  , u ,  ; n )} / ( I Z Z  J Z Z ) f 6 ( x; b )  p

Model Experiment on Heel-Induced Hydrodynamic Forces 395 f7 ( x;b)  [m x z H (u , v )u r  K v (u , v ; n )v  K v W (G /  , u , v ,  )v  K r (u , v ,; n )r  K r W (G /  , u , v ,  )r  K M N L (u , v , r )  K P (u ) p  K R N L (u ,  )  K  (u ; n )

(26)

 K w (G /  , u ,  ; n )  m gGZ ( )] / ( I xx  J xx )

(27)

f 8 (x;b)  {  K R (    C )  K R TD r} / TE

Here the underlined parts indicate the higher order terms added to the original model (Umeda, 1999). The comparison in the case that the ship experiences a periodic motion is shown in Fig. 16. No significant difference is found but the average of yaw angle is different to some extent. The comparison in the case that the ship suffers surf-riding, broaching and capsizing is shown in Fig. 17. The difference can be only found in the final stage of capsizing. The comparison in boundaries of ship motion modes for control parameters is shown in Fig. 18. Although the region of stable surf-riding decreases significantly and the region of broaching appears at high speed zone in the proposed model, it can reasonably predict broaching and capsizing around the critical Froude number where surf-riding occurs. From these comparisons, it could be concluded that the proposed model has sufficient accuracy for practical use but with less experimental efforts as compared to the enhanced model. Pitch(degrees)

20 10 0 0 -10 -20

60 40 20 0 -200 -40 -60

5

Pitch(degrees)

10

15

20

t(s) Roll(degrees)

5

10

15

20

t(s)

Yaw(degrees)

60 40 20 0 -200 -40 -60

5

10

15

20

t(s)

5

60 40 20 0 -200 -40 -60

5

10

15

20

t(s) Roll(degrees)

5

10

15

20

t(s)

Yaw(degrees)

Rudder(degrees)

45 30 15 0 -150 -30 -45

20 10 0 0 -10 -20

60 40 20 0 -200 -40 -60

5

10

15

20

t(s)

Rudder(degrees)

10

15

20

t(s)

45 30 15 0 -150 -30 -45

5

10

15

20

t(s)

Experiment Calculation

Fig. 16 Comparison of numerical results with H/=1/10, /L=1.637, Fn=0.3, c=-30 degrees. (left: enhanced model, right: proposed model)

396 H. Hashimoto et al. Pitch(degrees)

20 10 0 0 -10 -20 90 60 30 0 -300 -60 -90

5

Pitch(degrees)

10

15

20

t(s) Roll(degrees)

5

10

15

20

t(s)

Yaw(degrees)

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7 Conclusions Nonliear heel-induced hydrodynamic forces in severe following waves were measured by a new experimental method. Then the mathematical model taking these forces into account is developed, and is compared with the existing free running model experiment. As a result, the following conclusions are obtained:

Model Experiment on Heel-Induced Hydrodynamic Forces 397

1. 2. 3. 4. 5.

The proposed experimental method is applicable to measure heel-induced hydrodynamic forces for large heel angle even in severe waves. Heel-induced hydrodynamic surge force in waves is small and can be neglected for capsizing prediction. Heel-induced hydrodynamic sway force, yaw moment and roll moment in waves have certain nonlinearity with respect to roll angle especially over 20 degrees of heel angle. Heel-induced hydrodynamic forces in waves are important for accurate prediction of capsizing. Quantitative prediction of broaching and capsizing in severe following and quartering waves can be realised when the heel-induced hydrodynamic forces in waves as well as other higher order terms are consistently included.

Furthermore, the simplified model with taking only essential higher order terms into account is proposed, and it is confirmed that the proposed model has sufficient prediction accuracy of broaching and capsizing in following and quartering waves for a practical purpose.

Acknowledgements This research was supported by a Grant-in Aid for Scientific Research of the Ministry of Education, Culture, Sports, Science and Technology of Japan (No. 15360465) and a Fundamental Research Developing Association for Shipbuilding and Offshore of the Shipbuilders’ Association of Japan. The authors express their sincere gratitude to the above organisations.

References Grim O (1961) Beitrag zu dem Problem der Sicherheit des Schiffes im Seegang. Shiff und Hafen 6: 490-497 (in German) Hashimoto H, Umeda N, Matsuda A (2004) Importance of Several Nonlinear Factors on Broaching Prediction. J of Marine Science and Technol 9:2: 80-93 Nakato M and Kohara S (1982) Stability Experime\ts in the Following Sea with Ship Speed –An Utilization of Circulating Water Channel–. Proceedings of the 2nd Int Conf on Stab and Ocean Veh 399-410 Renilson MR, Manwarring T (2000) An Investigation into Roll/Yaw Coupling and Its Effect on Vessel Motions in Following and Quartering Seas. Proceedings of the 7th Int Conf on Stab and Ocean Veh 452-459 Umeda N (1999) Nonlinear dynamics of ship capsizing due to broaching in following and quartering seas. J of Marine Science and Technol 4:1: 16-26 Umeda N, Hashimoto H (2002) Qualitative Aspects of Nonlinear Ship Motions in Following and Quartering Seas with High Forward Velocity. J of Marine Science and Technol 6:2 111-121

398 H. Hashimoto et al. Umeda N, Hashimoto H, Matsuda A (2002) Broaching Prediction in the Light of an Enhanced Mathematical Model with Higher Order Terms Taken into Account. J of Marine Science and Technol 7:3: 145-155 Umeda N, Matsuda A, Hamamoto M et al. (1999) Stability Assessment for Intact Ships in the Light of Model Experiments. J of Marine Science and Technol 4:2: 45-57 Umead N, Renilson MR (1992) Broaching -a dynamic behaviour of a vessel in following seas-. Wilson PA (ed) Manoeuvring and control of marine craft, Computational Mechanics Publications 533-543 Umeda N, Renilson MR (2001) Benchmark Testing of Numerical Prediction on Capsizing of Intact Stability in Following and Quartering Seas. Proceedings of the 5th Int Stab Workshop 6.1.1-6.1.10

Perceptions of Broaching-To: Discovering The Past K.J. Spyrou School of Naval Architecture and Marine Engineering, National Technical University of Athens, 9 Iroon Polytechneiou, Zographos, Athens 15780, Greece

Abstract

The article summarizes the key findings of an investigation into old bibliographical sources, aiming to determine how broaching-to was considered in the past; especially in terms of the magnitude of the problem and its perception in the older days. Several relevant citations have been identified: in voyager’s records, in nautical journals, in training manuals, in encyclopaedias and even in literary sources. The fear of broaching-to is referred-to in a famous old British poem, inspired by a real 18th century disaster.

From Falconer (1804)

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_22, © Springer Science+Business Media B.V. 2011

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1 Introduction Whilst an essential grasp of the scientific basis of the broaching-to behaviour of ships has been achieved only after the middle of the 20th century, the subject has been alive in the nautical bibliography since much longer. In the Merriam-Webster dictionary, next to the definition of the word broach [“to veer or yaw dangerously so as to lie broadside to the waves – often used with to”], one is informed that, use of the term in a nautical context is traced as far back as the year 1705. Indeed, in several book excerpts from the 19th, 18th and even the 17th century one can read captivating narratives of real occurrences of broaching-to; attempts of defining the phenomenon formally; and steering practices devised specifically for dealing with it in the “difficult hour”. A glimpse into the days when navigation was, in its own capacity, an adventure reveals that, the prevention of broaching-to was a central issue of ship operation. For a very long time, and more for the past than for our days, the maritime community have associated the term with one of the most dreadful conditions that a ship could be engaged during operation in a rough sea. Nonetheless, current professionals of the maritime field maintain limited recollection of the past of the subject beyond, say, a 50-year horizon. Vivid descriptions of relatively recent broaching-to occurrences can be found, for example, in the review article of Conolly (1972). It is a fair conjecture that, the broaching-to behaviour of sailing ships should have been influenced by phenomena that could not play any role for modern motor ships. Especially, the arrangement of sails, and the incurred significant destabilising wind forces, should have been very influential for turning, despite all efforts, a ship broadside to the weather. The combination of wind and waves should have placed more heavy demands upon the Master of the old days. From our investigation comes out that, broaching-to was mostly feared when a vessel was heading towards the shore; especially when being very near to it. Moreover, it was often discussed within a scenario of ship sailing at high speed, “rushing before the weather”. The current paper is an attempt to shed some light about how broaching-to was perceived in that, almost forgotten, era. And, to possibly trace some seeds of scientific thinking, let pre-mature, about the nature and the mechanics of the phenomenon.

2 Deeping Into the Past 2.1 William Dampier (1699) “A New Voyage Round the World” At the dusk of 17th century, Captain William Dampier describes his experiences of navigating around the globe:

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“... and besides, often very violent and fierce, so that a Ship with her sails loose, would be in danger to be over-set by them, or at least lose Masts or Yards, or have the Sails split, besides the Consternation that all Men must needs be in at such a time, especially if the Ship, by any unforeseen accident, should prove unruly, as by the mistake of the Man at Helm, or he that Conns, or by her broaching to (author’s note: it appears as “too” but it is corrected in an Errata Table that is included) against all endeavours, which often happens when a fierce gust comes; which though it does not last long, yet would do much damage in a short time, and tho’ all things should fall out well, yet the benefit of it would not compensate the danger:”

2.2 A Sailor Turned Famed Poet - William Falconer’s “The Shipwreck” [Robert Anderson (1795)] A celebrated poem written by a sailor, William Falconer, firstly published in 1762 under the title ”The shipwreck” describes quite movingly what should have been the poet’s stunning personal experience of broaching-to, leading to loss of men and the ship. The drama unfolded off the shore of Attica; at the very spot where, according to the myth, the ancient king of Athens Aegeus had jumped into the sea (henceforth giving his name to the Sea in front), disillusioned by a wrong sign alluding to the loss of his son, the mythical hero Theseus, in his battle against the Minotaur in Minoan Crete. William Falconer was a native of Scotland and “bred to the sea in which he spent the greatest part of his life”. Unable to steer clear of his destiny, he went down deep in the Mozambique Chanel with the frigate Aurora, in the winter of 1769 (Knight 1837). The true story and proceedings surrounding his poem come out from the following extract: “… he served on board the Britannia, a merchantman, bound from Alexandria to Venice, which touched at the Island of Candia (author’s note: this is today’s Crete), whence, proceeding on her voyage, she met with a violent storm, that drove her on the coast of Greece, where she suffered shipwreck near Cape Colonne (author’s note: nowadays Cape Sounion, in Attica), three only of the crew being left alive. The dangers which he really experienced he feelingly described in his next performance intituled The Shipwreck, a Poem, in three Cantos, by a Sailor, 4to. 1762. He inscribed it to the Duke of York and prefixed a chart of the ship’s way and a section of the ship itself, in order to render the poem completely intelligible.”

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From Canto III: “While shoreward now the bounding vessel flies, Full in her van St. George’s cliffs arise; High o’er the rest a pointed crag is seen, That hung projecting o’er a moffy green. Nearer and nearer now the danger grows, And all their skill relentless fates oppose. For, while more eastward they direct the prow, Enormous waves the quiv’ring deck o’erflow. While, as she wheels, unable to subdue Her sallies, still they dread her broaching-to.” Amazing is the truly complete definition of broaching-to, supplied in a footnote of Anderson’s (1795) book and found also in several later publications of the poem (it could not be confirmed that this definition had appeared also in the 1762 original publication of Falconer’s poem): “Broaching-to is a sudden and involuntary movement in navigation, wherein a ship, whilst scudding or sailing before the wind, unexpectedly turns her side to windward. It is generally occasioned by the difficulty of steering her, or by some disaster happening to the machinery of the helm.” Reference to broaching appears also earlier in the poem, however what is implied there is the placing of the ship beam to the wave. It is unclear if this is assumed to be voluntary or not: Canto II: Forbid it, Heaven, that, in this dreadful hour, I claim the dangerous reins of purblind power! But should we now resolve to bear away, Our hopless state can suffer no delay. Nor can we, thus berest of every fail, Attempt to steer obliquely on the gale. For then, if broaching sideward to the sea, Our dropsy’d ship may founder by the lee; No more obedient to the pilot’s power, Th’o’oerwhelming wave may soon her frame devour.

2.3 John Hamilton Moore (1791) “The Practical Navigator” A definition of “broach-to” appears in this 18th century book by J. Hamilton Moore, who portrays himself as “Teacher of Navigation, Hydrographer and Chart Seller”. Although more concise, his definition conveys an identical meaning:

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“Broach to: Is when a Ship, on a sudden, lays her Broadside to the Sea, and is dangerous in bad Weather.”

3 Broaching-To In The 19th Century 3.1 “The Life-boat” (1855), Extract from the Article “Management of Ships in a Surf and Broken Water” (Vol. II, No 18). (Author’s note: According to the Merriam-Webster Dictionary, surf – date 1685 – “is the swell of the sea which breaks upon the shore; and the foam, splash, and sound of breaking waves”). In this 1855 article one finds quite a lengthy discussion on operational practices for avoiding broaching-to and also about what is the cause of the phenomenon: “On the second point, running before a broken sea, an equal variety of management is observable, as practised on the coast, yet all alike intended to meet the one great risk of “broaching-to”, which nearly all agree in considering to be the greatest danger to which a boat can be exposed, and to be that which calls for the most skill and management to obviate it. As before observed, the greater number of skilful boatmen on the coast are in the habit of checking a boat’s way through the water or of backing her against a heavy sea on its approach. Their practice is to stop the boat’s way by backing their oars until the crest of the wave has struck the boat’s stem and passed her midship part, then to give way again, running in on the back of that wave, as far as they may be able to, then watching for the approach of the next, and repeating the same operation until they arrive at the beach, being careful, by steering with oars at the quarter or stern to keep the boat as far as possible, end on to the direction in which the sea is running. It must be here observed, that this management is so far varied according to the character of the boat; that in cobles, and other square sterned boats which have their bows better formed for meeting a sea than their sterns are, their position is reversed before entering the broken water, and they are taken in stern foremost and bow outwards, but the same principle being acted on о rowing back to meet each heavy wave instead of running from it. In a sailing boat this principle can only be so far acted on as to diminish the boat’s speed through water by taking her in under a very educed amount of sail, which is commonly one and by towing weights or instruments lade for the purpose. The advocates of this system of management have certainly reason on their side, in addition to experience; for as all acknowledge that there is greater danger in running before a broken sea than in going off against it, it is obvious that the more the former operation be assimilated to the latterm the safer it must be also, which is therefore effected by an alternate progressive and retrogressive movement, the

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latter being effected at the particular moments when the progressive motion would be dangerous. The true theory on which this practice is founded we will endeavour presently to explain. Proceeding then to the opposite practice of giving a boat speed, in fact running away from the sea, which constitutes the other important distinction in practice. The principle then acted on is to escape from the danger as fast as possible and other expedients are then resorted to prevent the risk of broaching-to. The most common of these is to trim the boat by the stern, by bringing all moveable weights aft (this supposes her stern to be outwards; if she were being taken in stern foremost, she would then be trimmed by the head). The force of the sea or wind on either quarter has then less power to beat it off, and cause the boat to broach-to, than it would have if the stern were light. Another expedient is to tow a pig of ballast, or a basket, or other instrument, which by its weight or hold on the water has the effect of a drag on the rear end of the boat, and prevents its being beat to leeward by the sea, thus keeping her end on to it. On the coast of Norfolk the following ingenious plan is commonly practised. The boatmen there employ an instrument for the above purpose called a “drogue”; it is a conical shaped canvas bag, of the form of a common candle extinguisher, about 2 feet diameter at the base or mouth of the bag, and 6 feet long, having a small opening at the other end or apex of the cone When running before a heavy sea in broken water, the drogue is thrown over from the stern, and towed by a stout rope with the large end foremost, when it instantly fills, and from the resistance it opposes to the water, holds the stern back and prevents the boat’s broaching-to: as soon as the danger is past the large tow rope is let go, and the drogue then towed with the smaller end foremost by a small line attached to that end, it then immediately collapses is emptied of water and offers but little resistance. The steering with an oar on each quarter is another expedient, employed to prevent broaching-to, as when running, a boat will not answer her helm on being overtaken by a sea. The recommendation to watch for an opportunity and avoid a sea, equally in running as on going off, could only be practised at those localities where the beach is steep. In reply to the question, as to whether any particular kinds of boats are more liable than others to broach-to, the answers given are so conflicting and contradictory as to afford no information on the point. In reply to another question as to the cause of a boat’s broaching-to, the almost invariable answer is, “because the stern is thrown out of the water and the rudder therefore ceases to act”. From our own observation we have formed the opinion that this is not the case, although it is quite true that at the moment of broachingto, a bout will not answer her helm. The phenomenon of broaching-to, we believe, may be correctly accounted for as follows: - on a boat encountering a heavy broken sea or roller end on, if she be stationary or is being profiled in a contrary direction to the wave, she will receive its blow, and it will quickly pass by her, her own inertia preventing her being

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carried away by it. If however she is being propelled in the same direction as the waves, and running rapidly through the water with her stern towards them, on a wave overtaking her, its first effect is to throw her stern up and to depress her bow, but so far from her rudder being out of water, both it and her stern are buried in the crest of the wave; in consequence, however, of her previous motion being in the same direction as that of the wave, she now offers so slight resistance to it, that instead of its passing her, she is hurried along with it at a rapid rate over the ground, her stem high up still immersed in the crest of the sea, and her bow low down at its base; as the wave approaches shoaler water, its inshore surface approaches more and more nearly to a perpendicular, and the tendency of the boat to run down this steep inclination added to the momentum she has already from her previous motion, causes her to run her bow under water, when her buoyancy at that end being destroyed her stern still light is pressed onward by the summit of the wave, and the undercurrent from the last receding wave at the same time acting on her bow, she is instantly, if a short boat, turned “end over end,” or if a long one, capsized quarter wise If she have so high a bow that it does not become altogether immersed, or if, as in a life-boat, the end of the boat is occupied by a water-tight air-case to the height of the gunwale, so as to prevent the admission of the water over the bow, the effect then is that the boat is instantaneously turned round broadside to the sea, when again, unless she be a life boat of a superior description, she is almost certain to be upset. In the circumstances thus described, the sole cause of a boat’s running herself under water or broaching-to is that of running from a sea instead of awaiting it, and suffering it to pass by; and the cause of the rudder being useless to keep the boat end on to the wave, is not that it is thrown out of the water, (although at other times it doubtless is so), for it is actually buried in it as is also the stern of the boat up to her gunwale, but it is because it is stationary in it the crest of the wave having acquired an actually progressive motion equal to that of the boat. If on the other hand the wave passes the boat, as its crest advances from the stern to the fore part, the rudder and stern are thrown out of the water; steering oars are therefore a most valuable auxiliary aid when running before a sea, but we would recommend the use of a rudder as well. We have been rather prolix in our account of the phenomenon of broaching-to, because it is a very interesting one, which it is important should be understood in order to arrive at the proper management and to obviate its disastrous effects, which have been more fatal to the lives and property of boatmen on our coasts than those proceeding from any other cause whatever.” Further exchanges appear in subsequent issues of the same journal. In one case (Vol II, No 19) an interesting observation is presented about frequent occurrence of broaching-to events near the English coast town Deal, that is located to the NorthEast of Dover: “… nevertheless, we are positively informed that boats have been lost by broaching-to when running for the shore at Deal; and we have known a Suffolk yawl of 18 or 20 tons burden, broach-to, upset, and drown the greater part of her crew, when running under sail for the shore on as steep a beach as that at Deal.”

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3.2 Henry Coleman Folkard (1863) “The Sailing Boat” In the section “Causes of ship capsizing” of this book one finds a description of broaching. But more enlightening for the reader is the discussion that ensues, on the “drogues”. These were (and still are) used for helping the ship to keep her course and avoid broaching (and bow diving): “The action of the sea upon a boat running into a heavy surf, may be thus described: - when on the top of a heavy wave or roller, the bows are lifted high out of the water; then, as the sea recedes, the boat is hurled forward, and the bows are buried under water; when the sea acting powerfully on her head and fore gripe, twists her round, broadside to the waves, called “broaching to;” and the sea then runs over the gunwale into the boat; the next motion that inevitably follows, is a heavy lurch on the other side and another sea breaks completely over, and fills or capsizes the boat. This may happen either under sail or oars. There is considerable difficulty in preventing a boat from broaching to, when stem and stern are alternately lifted out of the water by the waves; and should the boat broach to and meet a very heavy roller, broadside on, the chances are fifty to one that she will be swamped. Drogues* are now a good deal used on the Eastern Coast, in both sailing and rowing boats; they serve to check the boat’s way, and keep her end on to the waves; and are, therefore, of great assistance to the crew, in preventing the boat from broaching to. Experience teaches, that when a heavy breaker follows the boat up astern, it is useless to attempt running away from it: then a question naturally arises, what must be done on the impulse of the moment. “For your lives men! back her astern; hard at it every one of you! and let the man in the stern-sheets creep forward a moment, to lighten the boat’s stern!” By this effort the wave strikes the boat kindly, and passes on; but if allowed to follow her up astern, so surely as such an experiment is tried, the sea will either curl over the stern, or the boat will broach-to and take it over the gunwale. * A drogue is a conical shaped collapsible bag, about two feet in diameter at the mouth, and four feet six inches in length. When towed by the mouth, the drogue fills with water, and draws heavily; thereby checking the progress of the boat. A tripping-line is made fast to the apex or pointed end, and by slacking the towingrope and hauling on the tripping-line, the drogue collapses, and may be drawn on board very easily.“ Even in our days, drogues [and indeed multiple (“series”) drogues] are considered as indispensable drag devices onboard sailing boats, intended to be used in heavy weather (consult for example Hinz 2003).

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3.3 John McNeill Boyd (1860) “A Manual for Naval Cadets”. Drawing upon his experience from the Royal Navy, Captain Boyd advises in his book young cadets of the time about broaching-to and how to avoid it through operation. In the first paragraph he is probably alluding to the condition of “surfriding”. His discussion on the drogues should also be noted. “The great danger when running before a broken sea, is that of broaching-to. The cause of a boat’s broaching-to when running before a broken sea or surf is, that her own motion being in the same direction as that of the sea, whether it be given by the force of oars or sails, or by the force of the sea itself, she opposes no resistance to it, but is carried before it. Thus if a boat be running with her bow to the shore and her stern to the sea, the first effect of a surf or roller on its overtaking her, is to throw up the stern, and as a consequence to depress the bow; if she then has sufficient inertia (which will be proportional to weight) to allow the sea to pass her, she will in succession pass through the descending, the horizontal, and the ascending positions, as the crest of the wave passes successively her stern, her midships, and her bow, in the reverse order in which the same positions occur to a boat propelled to seaward against a surf. This may be defined as the safe mode of running before a broken sea. But if a boat on being overtaken by a heavy surf, has not sufficient inertia to allow it to pass her, the first of the three positions above enumerated alone occurs, her stern is raised high in the air and the wave carries the boat before it, on its front, or unsafe side, sometimes with frightful velocity, the bow all the time deeply immersed in the hollow of the sea, where the water, being stationary or comparatively so, offers a resistance, whilst the crest of the sea, having the actual motion which causes it to break, forces onward the stern, or rear end of the boat. A boat will in this position sometimes, aided by careful oar steerage, run a considerable distance until the wave has broken and expended itself. But it will often happen that, if the bow be low it will be driven under water, when the buoyancy being lost forward, whilst the sea presses on the stern, the boat will be thrown (as it is termed) end over end; or if the bow be high, or it be protected, as in some life-boats, by a bow air chamber, so that it does not become submerged, that the resistance forward acting on one bow will slightly turn the boat’s head, and the force of the surf, being transferred to the opposite quarter, she will in a moment be turned round broadside by the sea and be thrown by it on her beam-ends, or altogether capsized. It is in this manner that most boats are upset in a surf, especially on flat coasts, and in this way many lives are annually lost amongst merchant seamen when attempting to land after being compelled to desert their vessels. Hence it follows that the management of a boat, when landing through a heavy surf, must as far as possible be assimilated to that when proceeding to seaward against one, at least, so far as to stop her progress shoreward at the moment of being overtaken by a heavy sea, and thus enabling it to pass her. There are different ways of effecting this object:-

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1st. By turning a boat’s head to the sea before entering the broken water, and then backing in stern foremost, pulling a few strokes ahead to meet each heavy sea and then again backing astern. If a sea be really heavy and a boat small, this plan will be generally the safest, as a boat can be kept more under command when the full force of the oars can be used against a heavy surf than by backing them only. 2nd. If rowing to shore with the stern to seaward, by backing all the oars on the approach of a heavy sea, and rowing ahead again as soon as it has passed to the bow of the boat, thus ,rowing in on the back of the wave; or, as is practised in some lifeboats, placing the after-oarsmen, with their faces forward, and making them row back at each sea on its approach. 3rd. If rowed in bow foremost, by towing astern a pig of ballast or large stone, or a large basket, or a canvass bag termed a “drogue” or drag, made for the purpose, the object of each being to hold the boat’s stern back and prevent her being turned broadside to the sea or broaching-to. Drogues are in common use by the boatmen on the Norfolk coast; they are conical-shaped bags of about the same form and proportionate length and breadth as a candle extinguisher about two feet wide at the mouth and four and a half feet long. They are towed with the mouth foremost by a stout rope or a small line termed a tripping-line, being fast to the apex or pointed end. Whеn towed with the mouth foremost they fill with water, and offer a considerable resistance, thereby holding back the stern; by letting go the stouter rope and retaining the smaller line, their position is reversed, when they collapse, and can be readily hauled into the boat. Drogues are chiefly used in sailing-boats, when they both serve to check a boat’s way and to keep her end on the sea. They are however a great source of safety in rowing-boats, and many rowing life-boats are now provided with them.”

4 Other References to Broaching-to With the opportunity of the establishment of the “Society for the Improvement of Naval Architecture” (forerunner of RINA) in London, Hutchinson (1791) advises that, in bad weather the best helmsman should be employed: “… and if the waves run high, when carrying a pressing sail, large, by such bad steering there is great danger of broaching the ship to; therefore none but the best helmsman should be permitted to steer at such times.” Later on, Hutchinson advises how to avoid broaching-to even when the foremast is lost. He refers to broaching-to in connection to scudding, a condition quite uniquely connected with sailing boats: The following entry appeared in one of the oldest editions of Encyclopædia Britannicca (1797): “Scudding, the movement by which a ship is carried precipitately before a tempest. As a ship flies with amazing rapidity through the water whenever this expedient is put in practice, it is never attempted in a contrary wind unless when her condition renders her incapable of sustaining the mutual effort of the wind and

Perceptions of Broaching-To: Discovering The Past 409

waves any longer on her side, without being exposed to the most imminent danger of being overset. A ship either feuds with a sail extended on her foremast, or, if the storm is excessive, without any sail; which in the sea phrase, is called scudding under bare poles. In sloops and schooners and other small vessels, the sail employed for this purpose is called the square sail. In large ships, it is either the foresail at large, reefed, or with its goose-wings extended, according to the degree of the tempest; or it is the fore-top sail, close reefed, and lowered on the cap; which last is particularly used when the sea runs to high as to becalm the foresail occasionally, a circumstance which exposes the ship to the danger of broaching-to. The principal hazards incident to scudding are generally a pooping sea; the difficulty of steering, which exposes the vessel perpetually to the risk of broaching-to; and the want of sufficient sea-room A sea striking the ship violently on the stern may dash it inwards, by which the must inevitably founder. In broaching-to (that is, inclining suddenly to windward) she is threatened with being immediately overturned; and, for want of sea-room, she is endangered by shipwreck on a lee-shore, a circumstance too dreadful to require explanation.” In another encyclopaedia was found a section offering advice for selecting the shape of a life-boat, with specific reference to the avoidance of broaching-to (Encyclopædia Perthensis 1816). We are informed also that, the invention of the lifeboat was claimed by William Wouldhave and that his proposed design was firstly built in 1789: “The curvature of the keel has however been demonstrated to be the principal or only error in the construction of the vessel and we would recommend to those who in future may construct such vessels to preserve the spheroidal form of the body of the boat, yet, so as to leave a straight keel and a sufficient quantity of gripe to hinder the boat from broaching to, on receiving the stroke of the waves on her ends.” In an American navigation textbook (Bowditch 1821) we learn about the difference between “broaching-to” and “bringing by the lee”: “To broach to. To incline suddenly to windward of the ship’s course, so as to present her side to the wind, and endanger her oversetting. The difference between broaching to and bringing by ht e lee may be thus defined: Suppose a ship under great sail is steering south, having the wind at N.N.W. then west is the weather side, and east the lee-side. If, by any accident, her head turns round to the westward, so that her sails are all taken a-back on the weather side, she is said to broach to. If, on the contrary, her head declines so far eastward as to lay her sails aback on that side which was the lee side, it is called bringing by the lee.” At the same time (and side of the Atlantic), Guest (1824) narrates his experiences of sea travel: “On the 25th, at 3, A.M. finding we could not, with any degree of safety, lay to any longer, we commenced scudding, under a close reefed foresail but soon found we could only scud under our bare poles, which we continued to do until 6 A.M.. The vessel then broached to, and lay with her broad side to the wind, in the trough of the sea (which is the most dangerous situation that a vessel can be in:) a most

410 K.J. Spyrou

tremendous sea was rolling down upon us, which we expected would soon swallow us up. I shall never forget the frightful looks and behaviour of the man who was at the helm when the vessel broached to. Although he was a very experienced seaman, he exclaimed calling on his God and Saviour -we are gone! we are gone! nothing can save us!” Fishbourne (1846) in his lectures on naval architecture argues for selecting a less fine stern compared to the bow: “The reason why short vessels do not run well.- It is notorious that vessels with fine after bodies, particularly if they be short, run badly. It is because there is so little action in the after body in such case, for the water cannot turn in upon it, there is even a danger in such vessels of their broaching to against their helm. The after body then should be greater than the fore body, in some ratio inverse of this action of the water.” In the shipbuilders guide of Partington (1826) we are advised that the use of sliding keels could save a ship: “… no misfortune, similar to that of broaching to, can ever befall a vessel furnished with sliding keels.” About these keels we learn from another passage referring to a 66 ft vessel: “ She has three sliding keels inclosed in a case or well; they are each 14 ft in length; the fore and the after keels are 3 ft broad each, and the middle keel is 6 feet broad. The keels are movable by means of a winch, and may be let down 7 feet below the real keel; and they work equally well in a storm as in still water.” Short references expressing the fear of broaching-to appear in numerous narratives of legendary expeditions around the globe. Two of them are mentioned below: 

In the search for a North-West passage from the Atlantic to the Pacific (Parry 1824).

 In the tales of the voyages of the ship H.M.S. Samarang in the islands of the Eastern Archipelago (Belcher 1848).

References Anderson R (1795) The Works of the British Poets: With Prefaces, Biographical and Critical, Printed for John & Arthur Arch; and for Bell & Bradfwte; and J. Mundell & Co. Edinburgh. Belcher E (1848) Narrative of the voyage of H.M.S. Samarang, during the years 1843-46; in the Islands of the Eastern Archipelago; Pub by Reeve, Benham, and Reeve, King William Street, Strand. Bowditch N(1821) The New American Practical Navigator. Published by E. M. Blunt, N. Y. McNeill Boyd J (1860) A Manual for Naval Cadets. Published by Longman, Green, Longman, and Roberts, 548 pp. Conolly JE (1972) Stability and control in waves: A survey of the problem. Journal of Mechanical Eng Science, Vol. 14, No 7. Dampier W (1699) A New Voyage Round the World, Published by J. Knapton. Encyclopædia Britannica: Or, A Dictionary of Arts, Sciences, and Miscellaneous Literature (1797), Edinburgh, Printed for A. Bell and C. MacFarquhar.

Perceptions of Broaching-To: Discovering The Past 411 Encyclopædia Perthensis; or Universal Dictionary of the Arts, Sciences, Literature &c. (1816) 2nd edition. Edinburgh, Printed by John Brown, Anchor Close. Falconer W (1804) The Shipwreck: a poem in three cantos. With a life of the author by James Stanier Clarke. Printed for William Miller, Old Bond Street, London. Fishbourne EG (1846) Lectures on Naval Architecture, Being the Substance of Those Delivered at the United Service Institution. Pub by John Russell Smith, London. Folkard HC (1863) The Sailing Boat: A Treatise on English and Foreign Boats , Descriptive of the Various Forms of Boats and Sails of Every Nation ; with Practical Directions for Sailing, Management &c , Pub from Longman, Green, Longman, and Roberts, 317 pp. Hinz E (2003) Heavy Weather Tactics Using Sea Anchors and Drogues, Paradise Cay Publications, 2nd Edition, 164 pages, ISBN-10 0939837374. Hutchinson W (1791) A treatise founded upon philosophical and rational principles: towards establishing fixed rules, for the best form and proportional dimensions in length, breadth and depth of merchsant ships in general; and also the management of them to the greatest advantage by pratical seamanship. Printed in Liverpool by Thomas Billinge, Castle Street. Knight C (1837) Penny Cyclopaedia of the Society for the Diffusion of Useful Knowledge, Vol. IX, Dionysius-Erne, Charles Knight and Co, 22 Ludgate Street, London. Moore HJ (1791) The Practical Navigator, and Seaman’s New Daily Assistant: Being an Epitome of Navigation: Including the Different Methods of Working the Lunar Observations. With Every Particular Requisite for Keeping a Complete Journal at Sea . Printed for and sold by R. Law and Son, London, 296 pages. Parry E (1824) Journal of a Second Voyage for the Discovery of a Northwest Passage from the Atlantic to the Pacific: Performed in the Years 1821- 22-23, in His Majesty’s Ships Fury and Hecla, Under the Orders of Captain William Edward Parry. Pub by J. Murray, London. The Life-boat, Or, Journal of the National Life-Boat Institution (1853). Pub by the Royal National Life-Boat Inst for the Preservation of Life from Shipwreck (Great Britain).

5 Nonlinear Dynamics and Ship Capsizing

Use of Lyapunov Exponents to Predict Chaotic Vessel Motions Leigh S. McCue* and Armin W. Troesch** *Aerospace and Ocean Engineering, Virginia Tech, **Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, Michigan

Abstract It is the aim of this paper to further the use of Lyapunov and local Lyapunov exponent methods for analyzing phenomena involving nonlinear vessel dynamics. Lyapunov exponents represent a means to measure the rate of convergence or divergence of nearby trajectories thus denoting chaos and possibly leading to the onset of conditions that produce capsize. The work developed here makes use of Lyapunov exponent methodologies to study capsize and chaotic behavior in vessels both experimentally and numerically using a multi-degree of freedom computational model. Since, the Lyapunov exponent is defined in the limit as time approaches infinity, one encounters fundamental difficulties using Lyapunov exponents on the capsize problem, which is inherently limited to a finite time. This work also incorporates the use of local Lyapunov exponents, which do not require an infinite time series, to demonstrate their usefulness in analyzing finite time chaotic vessel phenomena. The objective is to demonstrate the value of the Lyapunov exponent and local Lyapunov exponent as a predictive tool with which to indicate regions with crucial sensitivity to initial conditions. Through the intelligent use of Lyapunov exponents in vessel analysis to indicate specific regions of questionable stability, one may significantly reduce the volume of costly simulation and experimentation.

1 Background The method of Lyapunov characteristic exponents serves as a useful tool to quantify chaos. Specifically, Lyapunov exponents measure the rates of convergence or divergence of nearby trajectories (Haken, 1981; Wolf, 1986). Negative Lyapunov exponents indicate convergence, while positive Lyapunov exponents demonstrate

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_23, © Springer Science+Business Media B.V. 2011

415

416 L.S. McCue and A.W. Troesch

divergence and chaos. The magnitude of the Lyapunov exponent is an indicator of the time scale on which chaotic behavior can be predicted or transients decay for the positive and negative exponent cases respectively (Wolf, 1986). Physically, the Lyapunov exponent is a measure of how rapidly nearby trajectories converge or diverge. If one considers a ball of points in N-dimensional phase space, in which each point follows its own trajectory based upon the system’s equations of motion, over time, the ball of points will collapse to a single point, will stay a ball, or will become ellipsoid in shape (Glass & Mackey, 1988). The measure of the rate at which this infinitesimal ball collapses or expands is the Lyapunov exponent. For a system of equations written in state-space form x  u x  , small deviations from the trajectory can be expressed by the equation

    then defined by Equation 11. Often times only the maximal Lyaponov exponent is discussed since the maximal exponent is simplest to calculate from a numerical time series and yields the greatest insight into the dynamics of the system. However, for a space with dimension N, there are N Lyapunov exponents which make up the Lyapunov spectrum and correspond to the rate of expansion or contraction of the principal axes of the infinitesimal N-dimensional ball. For example, after ordering Lyapunov exponents with 1 being the largest and N being the smallest, the length of the most rapidly growing principal axis is proportional to e1t, the area of the two most rapidly growing principal axes is proportional to e1+2 t etc... (Wolf et al., 1985). Other useful quantities are the short time Lyapunov exponent and the local Lyapunov exponent. A short time Lyapunov exponent is simply a Lyapunov exponent defined over some finite time interval. The local Lyapunov exponent is a short time Lyapunov exponent in the limit where the time interval approaches zero. Both are dependent on starting points, and the short time Lyapunov exponent is also dependent on the magnitude of the time interval. Equations for short and local Lyapunov exponents are presented in Equations 2 and 3 respectively (Eckhardt & Yao, 1993). Since it is not practically possible in a numeric or experimental sense to take the limit T → 0, for the purposes of this paper the phrases ‘short time’ and ‘local’ Lyapunov exponents will be used interchangeably in reference to the approach given by Equation 2.

 xi   ui x   x j (Eckhardt & Yao, 1993). The maximal Lyapunov exponent is j

1The Lyapunov exponent is defined with the logarithm base e. Depending on the nature of the application, at times the exponent is calculated base 2 in order to allow the output to be expressed in terms of bits per second. This lends physical insight as to a rate at which information about the state of the system is created or destroyed (Schuster, 1984). For consistency, in this work, the exponent is always calculated using the true definition base e and is therefore expressed nominally with units of 1/time. ‘Nominally’ is due to the mixing of dimensions in the sections 3 and 4 of this work. It is important for the practitioner to give careful thought towards how they wish to treat multiple degree of freedom problems in calculation of Lyapunov exponents. Arguments can be made for maintaining dimensionality, and thus ‘weighting’ some degrees of freedom more heavily than others, while in other circumstancees it may make more sense to nondimensionalize all degrees of freedom in some consistent manner, such as nondimensionalizing all terms by the individual term’s standard deviation.

Use of Lyapunov Exponents to Predict Chaotic Vessel Motions 417

Experimental data was collected with a time step of 0.033 s and thus for consistency the short time Lyapunov exponent is calculated at this interval in the numerical model.

1 t  t

  lim log

xt  x0

T xt , x0 

(1)

xt  T  1 log T xt 

(2)

xt  T  xt 

(3)

1 T 0 T

local xt   lim log

One can draw conclusions about the nature of the dynamical system from the spectra assembled. For a one-dimensional system a positive Lyapunov exponent indicates chaos, a negative exponent defines a periodic orbit, and a zero value represents an orbit with marginal stability (Wolf et al., 1985). For the case of a three-dimensional system there are three Lyapunov exponents making up the spectra. A fixed point consists of all negative exponents. A limit cycle’s spectrum would have two negative values and a zero. A two-torus has two zeros and a negative value in its spectrum, and a strange attractor has one each of positive, negative, and zero values.(Wolf et al., 1985) This pattern can be extended to higher dimensional spaces. Since the Lyapunov spectrum is derived from a long-time evolution of an infinitesimal sphere, it is not a local quantity either spatially or temporally (Wolf et al., 1985). Therefore, one of the potential benefits to examining the capsize problem through the method described by Lyapunov exponents, is the number of numerical or physical experiments necessary to draw quantitative conclusions can be drastically reduced from the quantity which would be necessary by brute force simulation. The use of Lyapunov exponents to study capsize appears in the literature for both naval architecture and nonlinear dynamics. For example, the asymptotic Lyapunov exponent has been calculated from equations of motion for the mooring problem (Papoulias, 1987), single-degree-of-freedom capsize models (Falzarano, 1990; Murashige and Aihara, 1998a and Murashige and Aihara, 1998b; Murashige et al., 2000 and Arnold et al., 2003), work studying the effects of rudder angle while surf-riding as it leads to capsize (Spyrou, 1996), applied to parametric roll problems (McCue & Bulian, 2007, Neves et al., 2009), and real-time capsize warning for single and multi-DOF systems (McCue, 2004; McCue & Troesch, 2006; McCue et al, 2006; Story, 2009).

418 L.S. McCue and A.W. Troesch

The work of the following sections provides a detailed study into the use of Lyapunov exponents for analyzing stable, marginally unstable, and unstable dynamics leading to capsize. The goal is to evaluate Lyapunov exponents as a predictive tool with which to indicate regions of crucial sensitivity to initial conditions. To allow for detailed investigation into regions of questionable stability, one may reduce the necessary volume of simulations and experimentation of the entire phase space through intelligent use of Lyapunov exponents. Additionally, since the Lyapunov exponent provides a quantitative value, it can be used as a tool to directly compare the accuracy of a numerical model to experimental runs. Rather than attempting to qualitatively evaluate a numerical tool through predicting capsize with a heads or tails type comparison of capsize versus non-capsize, the Lyapunov exponent can be used to demonstrate that a numerical model is in fact simulating the same type, or magnitude of chaos as experiments.

2 Lyapunov Exponents from Experimental Time Series 2.1 Theory A large volume of work has been dedicated to the problem of calculating Lyapunov exponents from experimental time series. A number of researchers have developed methods which can be divided into two distinct approaches, direct methods and tangent space methods. Direct methods consist of searching the time series for neighbors at any given point and calculating expansion rates through comparison to these neighboring points. The first such method was that of Wolf et al (1985). Wolf et al (1985) developed a methodology in which one can calculate the largest positive Lyapunov exponent from a data set by following the long term evolution of one principal axis, a ‘fiducial trajectory’, progressively reorthonormalized maintaining phase space orientation (Wolf et al., 1985). Wolf’s method is highly sensitive to inputs, however, and can easily lead to an erroneous result. In the early 1990’s two separate research groups produced a new method (Rosenstein et al., 1993; Kantz, 1994). The approach eliminates the requirement Wolf imposes upon maintaining phase-space orientation stating it is unnecessary for calculating the largest Lyapunov exponent (Rosenstein et al., 1993). Additionally, rather than following one trajectory, the full data set is used, and in essence a trajectory for every pair of nearest neighbors is calculated. For details refer to Rosenstein et al (1993) and Kantz (1994). Both methods are substantively similar. The Kantz algorithm (and similarly the Rosenstein algorithm)

Use of Lyapunov Exponents to Predict Chaotic Vessel Motions 419

calculates the largest Lyapunov exponent by searching for all neighbors within a neighborhood of the reference trajectory and computes the average distance between neighbors and the reference trajectory as a function of time (or relative time scaled by the sampling rate of the data) (Kantz, 1994; Rosenstein et al., 1993). The algorithm computes values for Equation 4 with parameters defined as follows: xt, arbitrary point in time series; Ut, neighborhood of xt; xi neighbor of xt; , relative time scaled by sampling rate; T length of time series; S() stretching factor with region of robust linear increase showing slope equal to Lyapunov exponent ie et  eS() (Kantz, 1994; Kantz & Schreiber, 2004). However, this post-processing requirement of a robust linear increase in slope introduces new errors. While the method is useful and accurate for systems with known values for the Lyapunov exponent, the choice of region and parameters over which a ‘robust linear increase’ are found is somewhat arbitrary. It is the opinion of the authors that this tool is useful only if one knows what value of Lyapunov exponent is desired and can thus choose the region exhibiting a slope equal to that value.

S   

1 T  1  ln T t 1  U t



iU t

 xt   xi   

(4)

Tangent space methods, developed simultaneously by the separate research teams of Sano and Sawada (1985) and Eckmann and coauthors (1985; 1986) allow for calculation of the full spectrum of Lyapunov exponents through local predictions of the Jacobian along the time series trajectory. For example, for a given trajectory x(t) defined by Equation 5, the tangent vector  is given by the linearized form of Equation 5 presented in Equation 6 where J is the Jacobian matrix of f, J = f/x (Sano & Sawada, 1985). Sano and Sawada (1985) solve Equation 6 through a least squares estimate of the time dependent linear operator Aj which approximates the map from (0) to (t). The Lyapunov exponents are then computed using Equation 7 where  is a flow scale time increment, n is then number of data points, and e is an orthonormal basis maintained using a GramSchmidt renormalization process (Sano & Sawada, 1985). For details of this process refer to Sano and Sawada (1985) or the similar works of Eckmann et al (1985; 1986).

x  f x 

(5)

  J xt   

(6)

1 n  n

i  lim

n

 ln A e j 1

j j i

(7)

420 L.S. McCue and A.W. Troesch

The weakness of this approach is in its sensitivity to choice of embedding dimension. Too small an embedding dimension outputs erroneous Lyapunov exponents while too large an embedding dimension creates spurious exponents (Kantz, 1994). However, for this application with careful attention paid to the choice of embedding dimension, the tangent space method was found to be more robust then the direct methods as it was not dependent on any form of arbitrary postprocessing. To check for spurious exponents, the technique first suggested by Parlitz (1992) of analyzing both the original time series and the reversal of the original time series was used. No significant errors were noted for the largest Lyapunov exponents presented in this work. The implementation of the Sano and Sawada method included in the TISEAN (Hegger et al., 2000) package was used to calculate Lyapunov exponents for the experimental time series in the following subsections.

2.2 Application to the Capsize Problem The Sano and Sawada algorithm (1985) contained in TISEAN (Hegger et al., 2000) was applied to roll time series for the capsize and non-capsize experiments detailed in works by Obar and Lee (2001) and Lee et al (2006) with an embedding dimension equal to 6 to represent the six state-space variables. The physical model used in the experimental analysis and replicated numerically was a box barge featuring the following principal parameters: beam, B = 30.48 cm; beam/draft, B/T = 1.67; depth/draft, D/T = 1.06; beam/wave length, B/ = 0.23; wave amplitude/wave length, 0/= 0.01; roll natural frequency n = 2.28 rad/s; excitation frequency/roll natural frequency, e/n = 3.0; and angle of vanishing stability v = 11.4 degrees. The results of completing the process of using Sano and Sawada’s algorithm to calculate the largest Lyapunov exponent for all capsize and non-capsize runs are presented in Figure 1. As one might expect, there is relatively little variation in Lyapunov exponent values. The mean Lyapunov exponent for all runs is 1.76 s-1. The mean exponent for capsize runs, 1.83 s-1 is somewhat larger than that of noncapsize runs, 1.57 s-1 with significantly larger outliers. The histograms in Figure 2 show the distribution of Lyapunov exponents for both the capsize and non-capsize cases. Due to the brevity of the time series leading to capsize, it is impossible to confirm convergence of the Lyapunov exponent. However, it is feasible to discern an order of magnitude of the Lyapunov exponent with which to compare the numerical model as well as to conclude that runs leading to capsize are more chaotic in nature than those not leading to capsize as is apparent in Figures 1-2. The Lyapunov exponent is, by definition, an infinite time parameter, but by nature capsize is a finite event. For this reason greater insight can be gleaned through

Use of Lyapunov Exponents to Predict Chaotic Vessel Motions 421

calculation of the local Lyapunov exponent as will be discussed in Section 4. It should be noted that all runs result in a positive, chaotic, Lyapunov exponent. Recall that under the definition of the Lyapunov exponent a stable limit cycle would have a zero Lyapunov exponent. Therefore, the motions measured in the six dimensional state-space are growing rather that oscillating sinusoidally. Chaotic behavior is detected even in the non-capsize case.

3 Lyapunov Exponents from Ordinary Differential Equations 3.1 Theory As discussed in the previous section, the Lyapunov exponents measure the evolution of an infinitesimal sphere. However, infinitesimal quantities are not computationally feasible, and for a chaotic system with a finite initial separation between the principal axes of the sphere, it is impossible to assure convergence of solution for Lyapunov exponents before infinite values are encountered (Wolf,1986; Wolf et al., 1985). Therefore, other means of evaluation are necessary. To overcome this difficulty, Benettin et al proposed a method in which the trajectory of the center of the infinitesimal sphere, the fiducial trajectory, is defined by the nonlinear equations based upon initial conditions (Benettin et al., 1980). Principal axes are then calculated from the linearized form of the equations of motion about the fiducial trajectory (Wolf, 1986). These axes will, by definition, be infinitesimal relative to the attractor (Wolf et al., 1985). In implementation two computational difficulties arise, namely, computational limitations prohibit calculation of exponential growth andall basis vectors will have a tendency towards the direction of most rapid growth allowing for calculation of only the largest Lyapunov exponent (Wolf et al., 1985). To overcome these computational difficulties a Gram-Schmidt orthonormalization procedure is used (Wolf et al., 1985; Bay, 1999). Through noting the magnitude of vectors prior to re-normalization, growth rates are calculated. Additionally, by maintaining an orthonormal basis all Lyapunov exponents can be calculated. The first vector will naturally tend towards the largest rate of growth, the second vector towards the second most rapid growth rate, and so on (Wolf et al., 1985). Through the use of a Gram-Schmidt reorthonormalization the rapidly growing axes can be renormalized to represent an orthonormal basis maintaining the volume’s phase-space orientation (Wolf, 1986; Wolf et al., 1985). For further details on this process refer to (Benettin et al., 1980; Lichtenberg & Lieberman, 1983; Parker & Chua, 1989; Seydel, 1988; Wolf, 1986; Wolf et al., 1985).

422 L.S. McCue and A.W. Troesch

Fig. 1 Lyapunov exponents based on experimental time series. Phase space showing Lyapunov exponents versus roll and roll velocity initial conditions sampled at time t0. Model release time and length of time series variable between runs

Fig. 2 Histograms of Lyapunov exponent values for runs leading to capsize and non-capsize

Use of Lyapunov Exponents to Predict Chaotic Vessel Motions 423

3.2 Application to the Capsize Problem 3.2.1 Implementation Calculating the Jacobian of the equations of motion used in the numerical simulator (Obar et al., 2001; McCue & Troesch, 2003; Lee et al., 2006), given by Equation 8 is non-trivial. While the mass and linear damping terms are easily treated, the quadratic damping and forcing terms require extra consideration. m  a 22   0  a 42 

0

 xg  b22     yg    0  a 44    b42

a 24

0 m  a33

0

I cg

0 b33 0

b24  x g    0  y g  b1   

(8)

D  0 0 0  0   g e 2   f 2     D  0 0 0  0    g e 3   mg  f 3    0 0 b2      g e 4 GZ  f 4D   

Two approaches to treat the quadratic damping term are as follows. One method is to replace, in the linearized model, the term

  with  2 for   0 ,

  2 for   0 , and assume that the precise singularity at   0 will never be encountered due to double precision computational accuracy. The second approach is to use Dalzell’s (1978) treatment for quadratic damping. Dalzell

 k   (1978) fits an odd function series of the form     . Solving  k 1, 3,... k  k  2  c

5   35  3   over some c  for k the truncated third order fit becomes    16 48 c range  c    c (Dalzell, 1978). Basic testing indicated both treatments yield similar results, therefore the Dalzell treatment, with  = 10 degrees, was used for c

the results presented herein to avoid any difficulties due to the singularity associated with the first method. The linearized influence of the forcing side of the equation is calculated using a simple differencing scheme. Forces are calculated as the difference between their values on the fiducial trajectory and their values at the offset from the trajectory. Due to linear superposition this can be calculated in a more computationally

424 L.S. McCue and A.W. Troesch

efficient manner for the differential at (x+x, y+y, +, t) rather than conducting the summation of force differentials at (x+x, y, , t), (x, y+y, , t), and (x, y, +, t). Therefore, the linearized form of the equations of motion about the fiducial trajectory are written as Equation 9:

a24  xg  b22 0 b24     0  yg    0 b33 0 2   5 35     I  a44    b42 0 b1  b2 16 c  16 0 c   g e 2   f 2D   g e 2   f 2D        g e3  mg  f 3D    g e 3  mg  f 3D   g GZ  f D   g GZ  f D  e4 4 e4 4     m  a22  0   a42

0 m  a33



 x x , y y ,  ,t 

 x g     y g      



(9)

 x , y , ,t 

Six sets of these linearized equations are integrated in time to calculate the ith Lyapunov exponent for i=1,6 by measuring the logarithm of the rates of growth of the six systems. Numerically this corresponds to the discrete representation of Equation 1 as Equation 10 in which ‘m’ represents the number of renormalization steps conducted and ‘L’ denotes the length of each element. The methodology employed is derived from that published by Wolf and coauthors (Wolf et al., 1985; Wolf, 1986). m

L(t j 1 )

j 1

L(t j )

1 m  1t  log

(10)

3.2.2 Results Figure 3 presents a three-dimensional phase space portrayal with the value of the first Lyapunov exponent plotted on the z-axis. Ten minutes of simulated data was collected for non-capsize runs and capsize runs were simulated until the point of capsize. Initially this plot could appear enigmatic. Graphically, it is apparent that all non-capsize runs result in an equal valued positive Lyapunov exponent of approximately 1.77 s-1 after 600 simulated seconds. Intuitively it is expected that capsize runs would have larger Lyapunov exponent values than non-capsize, however capsize runs all feature small, even near-zero Lyapunov exponents.

Use of Lyapunov Exponents to Predict Chaotic Vessel Motions 425

Fig. 3. Lyapunov exponents based on numerical simulation. Phase space showing Lyapunov exponents versus roll and roll velocity initial conditions with numerical model released at t0

Fig. 4 Numerically calculated Lyapunov exponent as a function of time for neighboring capsize and non-capsize cases. Top panel shows full time series from t = 0 to t = 600. Bottom panel shows identical data over critical region from t = 25 to t = 40. Both runs released with zero initial roll velocity, sway, sway velocity, heave, or heave velocity. Initial roll for non-capsize and capsize runs equal to 0 and 1 degree respectively

426 L.S. McCue and A.W. Troesch

To make physical sense of this counter-intuitive result consider Figure 4 which shows the convergence, or lack thereof, of Lyapunov exponents for neighboring capsize and non-capsize cases. In Figure 4 it is evident that the time series resulting in capsize is too short to allow for convergence of the Lyapunov exponent. While the Lyapunov exponent of both the non-capsize and capsize cases follow closely, it is only immediately before capsize that the value of the Lyapunov exponent for the capsize case rises somewhat above the value of the Lyapunov exponent for the non-capsize case. Yet capsize occurs sufficiently rapidly as to prevent convergence of the exponent and instead results in small, non-converging, Lyapunov exponents. This explains the small values of Lyapunov exponents for capsize runs shown in Figure 3. As is evident, hundreds of cycles are necessary for exponent convergence. Even with a lack of convergence of the Lyapunov exponent for capsize cases, key information can be derived from the result. Primarily, since the Lyapunov exponent is an indicator of the magnitude of the chaos of the system, by way of comparison of experimental and numerical results for large amplitude motions near capsize this methodology serves as a validation tool for the numerical model. Consider Figures 1 and 3. In Section 2 it was shown that the Lyapunov exponent for experimental runs not leading to capsize was 1.57 s-1. This value is close to the Lyapunov exponent of approximately 1.77 s-1 for non-capsize runs using the numerical model. Additionally, due to the finite nature of the experimental tests, it is anticipated that if it were feasible to collect data over longer time interval, such as the ten minute interval over which the numerical model was simulated, the experimental Lyapunov exponent would converge to a somewhat larger value thus reducing the difference between the two mean exponents. It is impossible then to calculate an accurate value of the Lyapunov exponent for a capsize run. To understand the chaotic behavior leading to capsize it is important to consider a short term Lyapunov exponent instead as is discussed in Section 4.

4 Short Time Lyapunov Exponents from Ordinary Differential Equations 4.1 Implementation The short time Lyapunov exponent from ordinary differential equations is calculated in much the same manner as the Lyapunov exponent. Again, ‘n’ sets of differential equations linearized about the fiducial trajectory are calculated to measure incrementally stretching and shrinking principal axes. The ‘n’ linearized sets, where ‘n’ is the dimension of the phase space, are reorthonormalized after each step. Numerically, Equation 2 is calculated as Equation 11.

Use of Lyapunov Exponents to Predict Chaotic Vessel Motions 427

1 xt , t  

1 Lt  t  log t Lt 

(11)

4.2 Results Consider Figure 5 in comparison to Figure 4. In both figures the numerical model is released at to, the location of the maximum transient wave peak, or 27.1705 seconds (Lee et al., 2006). Figure 5 shows the short time Lyapunov exponent as a function of time for the same neighboring cases leading to capsize and noncapsize. The non-capsize case rises to a maximum value of 4.37 s-1 then, similarly to Figure 4, converges to oscillatory behavior about a short time Lyapunov exponent of 1.85 s-1. However, for the capsize case useful predictive information is now visible. For the last few steps prior to capsize the short time Lyapunov exponent is larger than the short time Lyapunov exponent for non-capsize although they initially feature similar behavior. At capsize the short time Lyapunov exponent rapidly increases to a value an order of magnitude larger than that of the noncapsize case.

Fig. 5 Numerically calculated short time Lyapunov exponent as a function of time for neighboring capsize and non-capsize cases. Top panel shows full time series from t = 0 to t = 600. Bottom panel shows identical data over critical region from t = 25 to t = 40. Both runs released with zero initial roll velocity, sway, sway velocity, heave, or heave velocity. Initial roll for non-capsize and capsize runs equal to 0 and 1 degree respectively

428 L.S. McCue and A.W. Troesch

Figure 6 presents the values of short time Lyapunov exponent as a function of initial roll and roll velocities. While Figure 3 featured relatively invariant values of Lyapunov exponent based upon capsize or non-capsize, Figure 6 yields a range of short time Lyapunov exponents for runs leading to capsize. In Figure 6 runs not leading to capsize all feature peak short time Lyapunov exponent values near 4.2 s-1 as shown by the histogram plot of the same data given in Figure 7. However, as seen in Figure 7 runs leading to capsize have a wide range of values for short time Lyapunov exponent with a mean of 8.3 s-1 and a standard deviation of 6.1 s-1, an order of magnitude larger than the standard deviation of the peak first short time Lyapunov exponent for runs not leading to capsize.

Fig. 6 Short time Lyapunov exponents based on numerical simulation. Phase space showing short time Lyapunov exponents versus roll and roll velocity initial conditions with numerical model released at to

It should be noted that peak short time Lyapunov exponent for capsize runs ranges from near zero to 33 s-1. It is the hypothesis of the authors that capsize runs featuring small short time Lyapunov exponents are a result of a ‘blue sky catastrophe’ or a process with hysteresis as discussed by Thompson and coauthors (1987) and Lee et al (2006). In such cases the rapid growth of the capsize attractor results in a capsize which is potentially difficult to detect unless the time interval of measurement is exceedingly small. This is consistent with the top panel of Figure 8 which indicates that the bulk of the runs with both the smallest and largest local Lyapunov exponents capsize within the first time steps immediately after the maximal exponent and the bottom panel which shows that the smallest

Use of Lyapunov Exponents to Predict Chaotic Vessel Motions 429

Fig. 7 Histograms indicating range of short time Lyapunov exponent values for runs leading to capsize and non-capsize

Fig. 8 (top) Peak value of largest local Lyapunov exponent as a function of the number of cycles from peak exponent to capsize. (bottom) Peak value of largest local Lyapunov exponent as a function of the number of cycles from release to capsize

Lyapunov exponents are found for runs which capsize within the first two cycles after release. Analysis of Figure 6 also indicates that that those capsize runs with

430 L.S. McCue and A.W. Troesch

smallest local Lyapunov exponent tend to be found at large negative initial roll angles. Many of these roll angles are statistically unlikely to occur and therefore choosing initial limits of phase space using a statistically validated methodology, such as that described in McCue and Troesch (2004; 2005) is a relevant first step in the analysis of real vessels operating in real seaways.

5 Conclusions From the study presented herein the usefulness of the Lyapunov exponent and short time Lyapunov exponent as analytical tools for studying vessel capsize is investigated. To summarize:  The Lyapunov exponent is a fundamental system parameter taking on fixed values in the non-capsize region with minimal transitional areas. It can be used to isolate regions of questionable stability to aid in the reduction of test matrices.  The Lyapunov exponent is shown effective as a validation tool for capsize simulators. Through consistency between the magnitude of a system parameter such as the Lyapunov exponent for large amplitude roll motions measured from an experimental time series and those measured from a numerical simulator, it is possible to demonstrate that the numerical model is capturing the inherent physics of the problem.  The inherent ineffectiveness of the Lyapunov exponent to yield informative results for the capsize problem is demonstrated. Rather, the importance of calculating a finite time value, such as a short time Lyapunov exponent is discussed.  From the calculation of the short time Lyapunov exponent further system information is gained which can be used as a predictive tool to indicate lost stability potentially leading to capsize. However, more work must be done to further identify the different types of capsize as characterized by small and large short time Lyapunov exponents. For more current work along these lines, the interested reader is referred to McCue and Troesch (2006) and Story (2009).

Acknowledgements The authors wish to express their gratitude for funding on this project from the Department of Naval Architecture and Marine Engineering at the University of Michigan and the National Defense Science and Engineering Graduate Fellowship program.

Use of Lyapunov Exponents to Predict Chaotic Vessel Motions 431

References Arnold L, Chueshov I & Ochs G (2003) Stability and capsizing of ships in random sea-a survey. Tech. rept. 464. Universität Bremen Institut für Dynamicsche Systeme Bay John S (1999) Fundamentals of Linear State Space Systems. McGraw-Hill Benettin G, Galgani L, Giorgilli A, Strelcyn JM (1980) Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: a method for computing all of them. Mecc. 9–20 Dalzell JF (1978) A Note on the Form of Ship Roll Damping. J of Ship Res, 22 3 178–185 Eckhardt B, Yao D (1993) Local Lyapunov exponents in chaotic systems. Phys D 65:100–108 Eckmann JP, Ruelle D (1985) Ergodic theory of chaos and strange attractors. Rev of Mod Phys 57 3: 617–656 Eckmann JP, Kamphorst S, Oliffson, Ruelle D, Ciliberto S (1986) Liapunov exponents from time series. Phys RevA 34 6 Falzarano JM (1990) Predicting complicated dynamics leading to vessel capsizing. Ph.D. thesis, Univ of Michigan Glass L, Mackey M (1988) From Clocks to Chaos. Princet, NJ: Princet Univ Press Haken H (ed) (1981) Chaos and Order in Nature. N.Y. Springer-Verlag Hegger R, Kantz H, Schreiber T et al (2000) TISEAN 2.1, Nonlinear time series analysis. http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.1/index.html Kantz H (1994) A robust method to estimate the maximal Lyapunov expon of a time series. Physi Lett A, 185:77–87 Kantz H, Schreiber T (2004) Nonlinear time series analysis. Second edn. Camb Univ Press Lee YW, McCue L, Obar M, Troesch A (2006) Experimental and numerical investigation into the effects of initial conditions on a three degree of freedom capsize model. J of Ship Res. 50:1 63-84 Lichtenberg AJ, Lieberman MA (1983) Regular and stochastic motion. N.Y.: SpringerVerlag McCue LS (2004) Chaotic vessel motions and capsize in beam seas. Ph.D. thesis, Univ of Michigan McCue LS, Bassler C, Belknap W (2006) Real-time identification of behavior leading to capsize. 9th Int Conf on Stab of Ships and Ocean Veh (STAB 2006), Rio J. Braz, Sept McCue LS, Bulian G (2007) A numerical feasibility study of a parametric roll advance warning system. J of Offshore Mech and Arct Eng, 129:3 165–175 McCue LS, Troesch AW (2003-sept). The effect of coupled heave/heave velocity or sway/sway velocity initial conditions on capsize modelling. In: 8th Int Conf on the Stab of Ships and Ocean Veh McCue LS, Troesch AW (2004-july). Overlay of probability density functions on multidegree of freedom integrity curves for statistical prediction of critical capsize wave height. Soc for Ind and Appled Math, Annual Meet, Portland, Oregon McCue LS, Troesch AW (2005) Probabilistic determination of critical wave height for a multi-degree of freedom capsize model. Ocean Eng. 32:13 1608–1622 McCue LS, Troesch AW (2006) A combined numerical-empirical method to calculate finite time Lyapunov exponents from experimental time series with application to vessel capsizing. Ocean Eng. 33:13 1796–1813 Murashige S, Aihara K (1998) Coexistence of periodic roll motion and chaotic one in a forced flooded ship. Int J of Bifurc and Chaos, 8:3 619–626 Neves MAS, Vivanco JEM, Rodríguez CA (2009) Nonlinear dynamics onparametric rolling of ships in head seas. 10th Int Conf on Stab of Ships and Ocean Veh., St. Petersburg, Russia, June 2009

432 L.S. McCue and A.W. Troesch Obar MS, Lee YW, Troesch AW (2001-Sept). An experimental investigation into the effects initial conditions and water on deck have on a three degree of freedom capsize model. In: Fifth Int Workshop on the Stab and Oper Saf of Ships Ott E, Sauer T, Yorke J (eds) (1994). Coping with Chaos. NY.: John Wiley and Sons Papoulias FA (1987). Dynamic analysis of mooring systems. Ph.D. thesis, Dep of Nav Archit and Marine Eng, Univ of Michigan, Ann Arbor, MI Parker TS, Chua LO (1989) Practical numerical algorithms for chaotic systems. NY. Springer-Verlag Parlitz U (1992) Identification of true and spurious Lyapunov exponents from time series. Int J of Bifurc and Chaos 2:155–165. Reprinted in (Ott et al., 1994) Rosenstein MT, Collins JJ, De Luca CJ (1993) A practical method for calculating largest Lyapunov exponents from small data sets. Phys D, 65:117–134 Sano M, Sawada Y (1985) Measurement of Lyapunov spectrum from a chaotic time series. Phys Rev Lett. 55 10 Schuster HG (1984) Deterministic Chaos. Ger: Phys-Verlag Seydel R (1988) From Equilibrium to Chaos. N.Y.: Elsevier Science Publ Spyrou K (1996) Homoclinic connections and period doublings of a ship advancing in quartering waves, Chaos 6 2 Story WR (2009) Application of Lyapunov exponents to strange attractors and intact & damaged ship stability. MS Thesis. Dep of Aerosp and Ocean Eng. Virginia Tech Blacksbg VA Thompson JMT, Bishop SR, Leung LM (1987). Fractal basins and chaotic bifurcations prior to escape from a potential well. Phys Lett A 121 3 Wolf A (1986) Quantifying chaos with Lyapunov exponents. Chaos. Princet, NJ: Princet Univ Press 13 Wolf A, Swift J, Swinney H, Vastano J (1985) Determining Lyapunov exponents from a time series. Phys D, 16 285–317

Applications of Finite-Time Lyapunov Exponents to the Study of Capsize in Beam Seas Leigh S. McCue* *Aerospace and Ocean Engineering, Virginia Tech

Abstract This paper demonstrates the use of finite-time Lyapunov exponents (FTLE) for the detection of large amplitude roll motions and capsize. The study is conducted for a simple, single degree of freedom model of the Edith Terkol in regular and random beam seas. The effectiveness of the FTLE technique is qualitatively compared to a scalar ship motion metric based upon the ‘energy index (EI)’ concept used for real-time identification of quiescence.

1 Introduction For centuries chaotic vessel motions and capsize have resulted in lost cargo, ships, and lives. For this reason it is of the utmost importance to develop tools which can provide mariners with indicators of inclement ship motions. Lyapunov exponents provide a measure of the rate of convergence or divergence of nearby trajectories thus indicating the level of sensitivity of a system to initial conditions. They have been used to demonstrate exponentially divergent, chaotic behavior in vessel motions as demonstrated by references such as Papoulias (1987), Falzarano (1990), Spyrou (1996), Murashige and collaborators (1998a; 1998b; 2000), Arnold et al. (2003), and McCue and Troesch (2004; 2004; 2006). The Lyapunov exponent, however, is defined in the limit as time approaches infinity; therefore to detect instabilities in time leading to an event, such as capsize, the finite-time form of the Lyapunov exponent is necessary. For a system of   ux  , small deviations from the equations written in state-space form x trajectory can be expressed by the equation Yao,1993).

x is

 

 j 

x i   u i x x j (Eckhardt

&

a vector representing the deviation from the trajectory with

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_24, © Springer Science+Business Media B.V. 2011

433

434 L.S. McCue

components for each state variable of the system. From this, the equation for the finite-time Lyapunov exponent can be written as Equation 1 (Eckhardt & Yao, 1993).

T xt , x0 

xt  T  1 log T xt 

(1)

For a multi-dimensional state space a spectrum of finite-time Lyapunov exponents can be calculated with the same number of exponents as dimension of the state-space. Methods for computing Lyapunov exponents and finite-time Lyapunov exponents from equations of motion are well developed in the literature with details available in references such as Benettin et al. (1980), Wolf et al. (1985), and Eckhardt and Yao (1993). In this work, the results generated via finite-time Lyapunov exponents are compared to results from a path-independent scalar ship motion metric known as the energy index (EI). The energy index, initially proposed by O’Reilly (1987), is an experimentally-tested, empirically-based indicator of quiescent periods suitable for sea-based aviation operations in which an energy-like scalar value is used as a measure of ship motions (O’Reilly, 1987; Ferrier & Manning, 1998; Ferrier et al., 2000a; Ferrier et al., 2000b). While it is desirable for current ship-air operations to indicate motions in all six degrees of freedom, particularly vertical accelerations (Colwell, 2002b), this study compares finite-time Lyapunov exponents to a simple formulation of the energy index based upon the two state space variables considered in this work, namely roll and roll velocity. The intention of this work is to demonstrate the capabilities of FTLE algorithms for detecting instabilities without the dependence more complicated energy index algorithms have upon empirically defined coefficients (Ferrier & Manning, 1998). It should be clear, however, that this is not the traditional use of the energy index, rather that the energy index concept in this case is being used as a benchmark. Because the energy index has been experimentally tested and shown to accurately indicate regions of quiescence (O’Reilly, 1987), it provides a useful comparison for this proof-of- concept study. This work aims to illustrate that a mathematical approach, such as the FTLE, can yield as much, or greater information into the fundamental physics driving the motions of the system to capsize or quiescence, than a method with ship and condition-dependent parameters.

Study of Capsize in Beam Seas 435

2 Regular Seas 2.1 Regular Seas: Modelling Details For the study presented in this work, the simple single degree of freedom roll model for the Edith Terkol established by Soliman and Thompson (1991) and given in Equation 2 was simulated.

  b1  b2    c1  c 2    c3 3  c 4   3  c5 5 

M t  WM  (2) I I

In Equation 2, bi and ci terms are linear and nonlinear damping and stiffness coefficients respectively with specific values: b1 = 0.0043, b2 = 0.0225, c1 = 0.384, c2 = 0.1296, c3 = 1.0368, c4 = -4.059, c5 = 2.4052, I = 1174, and WM = 0 (Soliman & Thompson, 1991). The term M(t) in regular seas is defined by Equation 3 (Senjanović et al., 2000), where 0, the effective wave slope, is taken to equal 0.73 (Francescutto & Serra, 2002; Gu, 2004), n and e equal 0.62 and 0.527 rad/s respectively (Soliman & Thompson, 1991), and the terms H = 4.94m and  = 221.94m are chosen to be consistent with results published by Soliman and Thompson in which interesting behavior is detected (1991). The general form of Equation 3 is widely available in the literature (Nayfeh & Khdeir, 1986; Nayfeh & Sanchez, 1990). Each simulation was allowed to run for 100 wave encounters or to the point of capsize. For the purposes of this work capsize was defined as exceeding the vessel’s angle of vanishing stability, v = 0.88 rad (Soliman & Thompson, 1991).

M t   I 0 n2

H



sin  e t

(3)

2.2 Regular Seas: Finite-Time Lyapunov Exponents Finite-time Lyapunov exponents were calculated via simulation for the numerical model of the Edith Terkol given by Equations 2–3. Particular attention was paid to the values and timing of maxima in the FTLE time series. Figure 1 presents histograms illustrating the values of maximum FTLE values for both capsize (top) and non-capsize (bottom) cases. Additionally, by consideration of the middle panel of Figure 1, it is apparent that those capsize cases with small FTLE values correspond to cases which capsize within one excitation cycle, i.e. those runs which capsize, for all practical purposes, immediately. Figure 2 illustrates that elimination of

436 L.S. McCue

those cases which capsize prior to one wave cycle essentially results in removal of the cases at the extrema of roll and roll velocity initial conditions.

Fig. 1 Histograms of maximum FTLE1 for capsize and non-capsize cases in regular seas

Fig. 2 Safe basins indicating non-capsize and capsize for all cases (left) and those cases capsizing after one excitation period (right) in regular seas

To demonstrate that instabilities are detectable in a time frame suitable for corrective measures to be taken, Figure 3 presents histograms of the number of wave cycles encountered between the occurrence of the maximum FTLE to the actual time of capsize. Again, the top panel includes all capsize runs while the bottom panel is restricted to those runs which capsize after one wave period. Considering only those runs which capsize after one encounter cycle, the majority capsize more

Study of Capsize in Beam Seas 437

than half a wave cycle after the maximum FTLE, with many capsizing multiple cycles after the location of the maxima. Detecting trends towards increasing values prior to the occurrence of maxima could lead to even greater time windows in which stabilizing controls changes are executed. This is discussed further in Section 2.3.

Fig. 3 Histograms of wave encounter cycles from the occurrence of the maximum FTLE1 to the time of capsize in regular seas

2.3 Regular Seas: Finite-Time Lyapunov Exponents in Comparison to the Energy Index For the single degree of freedom model, a simple definition of the energy index (EI) was used as given in Equation 4 (O’Reilly, 1987). Equation 4 emulates the form of the equation for the EI of O’Reilly’s for motions in multiple degrees of freedom (O’Reilly, 1987). More complicated EI models exist which rely upon empirically determined weighting coefficients for each degree of freedom (Ferrier & Manning, 1998; Ferrier et al., 2000a; Ferrier et al., 2000b). The purpose of this research is to demonstrate the ability to warn of large amplitude motions without a reliance upon empirical coefficients, thus justifying the use of the definition in Equation 4.

EI   2   2

(4)

438 L.S. McCue

Figure 4 presents a comparison of roll, FTLE, and EI values for a single ship motion simulation of the Edith Terkol with initial conditions (  rad,  rad/s) = (0, 0.2269). While the energy index reflects the instantaneous behavior of the system, the FTLE time series exhibits odd behavior, deviating from a fairly regular periodic response, 20 seconds prior to capsize and encounters peak values one cycle before capsize. Of particular interest is the middle plot, which displays the sum of the two FTLE for an entropy rate-like measure (Falkovich & Fouxon, 2004). Specifically, one cycle prior to capsize, this value rises above zero. Similar behavior is identified when examining a range of capsize and non-capsize cases in Figure 5 which present simulations near to the initial conditions of Figure 4. Those cases leading to capsize begin to exhibit unique FTLE time series behavior multiple cycles prior to capsize in addition to phase differences and peaks relative to the non-capsize runs. Additionally, phase differences in the FTLE time series are present in the non-capsize runs as time progresses. Some runs experience increasing roll motions while others encounter decreasing motions. Based upon this model, there is promise that measurement of these differences and detection of peaks can indicate variations in magnitude of ship motion in addition to capsize. Quantitatively, consideration of the sum of the FTLE values for those neighboring capsize and non-capsize cases in Figure 5 indicates that all capsize cases have positive FTLE sums and non-capsize cases have negative FTLE sums. A summary of initial conditions and values of peak FTLE1+FTLE2 displayed in Figure 5

is as follows: (  rad,  rad/s, FTLE1+FTLE2 1/s) = (0, -0.2618, 0.0025), (0, -0.2443, 0.0012), (0, -0.2269, 5.1439e-004), (0, -0.2094, -0.0030), (0, -0.1920, -0.0041), (0, 0.1745, -0.0049).

Fig. 4 Roll plotted with first and second FTLE (top), sum of FTLE (middle), and EI (bottom) for Edith Terkol in regular beam seas,



= 0 rad,



= -0.2269 rad/s

Study of Capsize in Beam Seas 439

Fig. 5 FTLE1 (top), sum of FTLE (middle top), EI (middle bottom), and roll (bottom) for Edith

 = 0 rad,  =-0.2618 rad/s;  = 0 rad,  = -0.2443 rad/s;  = 0 rad,  = -0.2269 rad/s;  = 0 rad,  = -0.2094 rad/s;  = 0 rad,  = -0.1920 rad/s;  = 0 rad,  = -0.1745 rad/s. Dotted and Terkol in regular beam seas. Six cases shown corresponding to initial conditions:

solid lines denote capsize and non-capsize runs respectively

Figure 6 shows the range of values for maximum FTLE1 + FTLE2 for non-capsize cases and those capsize cases in which capsize occurs after one excitation period. The mean value for capsize cases is 0.00098 (1/s) while the mean value fornoncapsize cases is -0.0037 (1/s). Positivity of the entropy rate-like value FTLE1 + FTLE2 is an indicator of potential danger. As apparent in Figure 7, those noncapsize runs with positive values for peak FTLE1 + FTLE2 represent simulations in the regions of greatest sensitivity to initial conditions. Thus, in addition to those areas resulting in capsize, it is not illogical to identify such conditions as necessitating precautionary measures and that a definition based upon FTLE methods could provide a conservative indicator of capsize.

440 L.S. McCue

Fig. 6 Histograms of maximum FTLE1 + FTLE2 for capsize and non-capsize cases in regular seas

Fig. 7 3-dimensional plot of initial conditions and maximum FTLE1 + FTLE2 for capsize and non-capsize cases in regular seas

Study of Capsize in Beam Seas 441

3 Random Seas 3.1 Random Seas: Modelling Details In random seas, the term M(t) in Equation 2 is written as Equation 5 (Senjanović et al., 2000; Gu, 2004):

M t   I 0 n2

2 d g

N

 i 1

2 i

S i  sin i t   i 

(5)

where N = 180 is the number of wave components, d = 0.0083 rad/s is the size of the wave frequency increment, i is the encounter frequency for each wave component, i is a random phase angle between 0 and 2π for each wave component, and S() is a wave energy spectrum. For this study the ISSC two-parameter spectrum, Equation 6 was used with a significant wave height Hs= 4.94m and characteristic frequency z = 0.527 rad/s for consistency with Section 2.



     

   S    0.11H s2 z5 exp  0.44 z   4





4

(6)

Fig. 8 Safe basin indicating non-capsize and capsize in random seas for those cases capsizing after one excitation period

442 L.S. McCue

Details as to the derivation of Equation 5 can be found in Senjanović et al. (2000), Jiang et al. (1996), and Chapter VIII of Principles of Naval Architecture (1989). A sample safe basin representing the simulation of Equations 2 and 5 is presented in Figure 8. In Figure 8 the same random seed is used for each simulation, therefore each initial condition pair is subject to the same random wave time series.

3.2 Random Seas: Finite-Time Lyapunov Exponents in Comparison to the Energy Index In random seas, finite-time Lyapunov exponents and energy index values were calculated via simulation for the numerical model of the Edith Terkol given by Equations 2 and 5. Figure 9 compares data for two time series with identical initial conditions in random seas in which one case leads to capsize, and one non-capsize. As in Figures 4-5, EI values serve as a reflection of the roll time series behavior. Conversely, the finite-time Lyapunov exponent time series demonstrates distinct variations in behavior well before the point of capsize.

Fig. 9 FTLE1 (top), sum of FTLE (middle top), EI (middle bottom), and roll (bottom) for Edith Terkol in random beam seas. Both cases correspond to initial conditions



= 0 rad,



= 0 rad/s.

Dotted and solid lines denote capsize and non-capsize respectively

For example, after approximately 540 seconds of simulation, there is a marked phase change between capsize and non-capsize FTLE time series in which the run leading to an ultimate state of capsize changes from leading to lagging the noncapsize time series. Additionally, the FTLE1 values increase substantially at

Study of Capsize in Beam Seas 443

approximately 550 seconds of simulation. In consideration of the sum of the two FTLE for both capsize and non-capsize runs, it is found that over the entire simulation time the maximum value for FTLE1 + FTLE2 for the capsize case is positive, 0.0028 while it is -0.0028 for the non-capsize case. This again lends validity to the belief that quantitative indicators of capsize can be formulated via the use of the FTLE as well as that the FTLE holds promise as an indicator of periods of coming quiescence. Figure 10 shows histograms of the maximum first FTLE for capsize and noncapsize cases in random seas. Only those runs capsizing after one excitation cycle are considered as established in Section 2. In Figure 10, similar to Figure 1, those runs resulting in capsize typically experienced larger maximum FTLE. Also shown in Figure 10 is the cycles from the occurrence of the peak FTLE to capsize. While in random seas, the cycles to capsize is dependent upon the random seed in the phase term of the wave train, it is apparent for the sample set presented herein, that for many cases capsize occurs sufficiently long after the peak in the FTLE time series that corrective measures could be implemented.

Fig. 10 Histograms of maximum FTLE1 for capsize (top) and non-capsize (middle) cases in random seas as well as cycles from maximum FTLE1 to capsize (bottom)

Figures 11–12 plot histograms and state space of the entropy rate-like value summing the two finite-time Lyapunov exponents. Those runs which capsize after one excitation cycle have, on average, a positive maximum FTLE1 +FTLE2 while those runs which do not capsize tend toward negative values as shown in Figure 11. As seen in Figure 12 those non-capsize cases with positive maximum FTLE1 + FTLE2

444 L.S. McCue

often lie on the fractal boundaries between capsize and non-capsize. Measures detecting positive, or small negative values of the sum of all Lyapunov exponents, in addition to detecting peaks, phase differences, and other oddities in the FTLE time series, can indicate impending capsize.

Fig. 11 Histograms of maximum FTLE1 +FTLE2 for capsize and non-capsize cases in random seas

Fig. 12 3-dimensional plot of initial conditions and maximum FTLE1 + FTLE2 for capsize and non-capsize cases in random seas

Study of Capsize in Beam Seas 445

4 Conclusions Based upon numerical simulation of Soliman and Thompson’s (1991) simplified single degree of freedom roll motion model, this work demonstrates that finitetime Lyapunov exponent calculations provide viable indicators of impending capsize in both regular and random seas. Instabilities leading to capsize are detected through: 1. peaks in the FTLE time series, 2. time series phase changes and/or other qualitative idiosyncratic behavior, and 3. positivity of the sum of the FTLE spectrum in time. This further supports the conclusions arrived at by McCue and Troesch in their studies of three-degree of freedom numerical and experimental results (2004; 2006). To implement this theory in a practical, on-board manner, a number of areas for future research arise. Specifically, it would be of use to further study the feasibility of this technique in detecting quiescence as applied to ship based aviation applications via comparison of results from both methods for ship motions in six degrees of freedom. Colwell expressed concern that a scalar representation of the ship motion environment obscures some of the understanding of the limiting aspects of sea-air operations (2002a). It is the author’s belief that a mathematical approach, such as that presented by finite-time Lyapunov exponents, enables understanding of the fundamental physics of the system while still reaping the benefits of a scalar output. Additionally, more quantitative means to detect variations in the FTLE time series are required. It is not always practical, feasible, or computationally efficient to calculate the entire FTLE spectrum, and thus measuring for positivity of the sum of the finite-time exponents may not be the ideal quantitative indicator. Instead, benchmarks for peak values and techniques to quantify phase and other behavior changes in the FTLE1 time series must be developed through further simulation and analysis of experimental data for a range of hull forms in realistic seas. For further research along these lines the interested reader is referred to Story (2009).

Acknowledgements This research was conducted while funded by the ONR-ASEE Summer Faculty Research Program. Additionally, the author wishes to thank William Belknap (Seakeeping Division, Code 5500, NSWC-CD), Judah Milgram (Sea-Based Aviation, Code 5301, NSWC-CD), and Armin Troesch (University of Michigan) for their assistance and insights on this topic.

446 L.S. McCue

References Arnold L, Chueshov I and Ochs G (2003) Stability and capsizing of ships in random sea-a survey. Tech rept 464. Universität Bremen Institut für Dynamicsche Systeme Beck RF, Cummins WE, Dalzell JF, Mandel P and Webster WC (1989) Motions in Waves. Chap. VIII of: Lewis EV. (ed) Princip of Nav Archit Soc of Nav Archit and Mar Eng Benettin G, Galgani L, Giorgilli A and Strelcyn JM (1980) Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: a method for computing all of them. Mecc 9-20 Colwell JL (2002-a May) Maritime Helicopter Ship Motion Criteria-Challenges for Operational Guidance. In: NATO RTO SCI-120 Symp on \Chall.... Colwell JL (2002-b) Real time ship motion criteria for maritime helicopter operations. In: ICAS 2002 Congr Eckhardt B, Yao D (1993) Local Lyapunov exponents in chaotic systems. Phys D 65: 100-108 Falkovich G and Fouxon A (2004) Entropy production and extraction in dynamical systems and turbulence. New J of Phys 6 50 Falzarano JM (1990) Predicting complicated dynamics leading to vessel capsizing. Ph.D. thesis, Univ of Michigan Ferrier B and Manning T (1998) Simulation and testing of the landing period designator (LPD) helicopter recovery aid. Nav Eng J 110 1: 189-205 Ferrier B, Baitis A E and Manning A (2000a) Evolution of the Landing Period Designator (LPD) for Shipboard Air Ope. Nav Eng J 112 4 297-315 Ferrier B, Applebee T, Manning A and James D (2000b-May) Landing period designator visual helicopter recovery aide: theory and real-time application. In: Proc of the 56th Annual Forum of the American Helicopter Soc Francescutto A and Serra A (2002) Experimental tests on ships with large values of B/T, OG/T and roll period. In: 6th Int Ship Stab Workshop Gu JY (2004) Nonlinear rolling motion of ship in random beam seas. J of Marine Sci and Technol, 12(4): 273-279. Nat Taiwan Ocean Univ, publ Jiang C, Troesch AW and Shaw SW (1996) Highly nonlinear rolling motion of biased ships in random beam seas. J of Ship Res 40 2: 125-135 McCue LS (2004) Chaotic vessel motions and capsize in beam seas. Ph.D. thesis, Univ of Michigan McCue LS and Troesch AW (2004-Nov) Use of Lyapunov exponents to predict chaotic vessel motions. In: 7th Int Ship Stab Workshop McCue LS and Troesch AW (2006) A combined numerical-empirical method to calculate finite time Lyapunov exponents from experimental time series with application to vessel capsizing. Ocean Eng. 33 Issue 13:1796-1813 Murashige S, Yamada T and Aihara K (2000) Nonlinear analyses of roll motion of a flooded ship in waves. Philos Transactions of the Royal Soc of Lond A 358:1793-1812 Murashige S and Aihara K (1998a) Co-existence of periodic roll motion and chaotic one in a forced flooded ship. Int J of Bifurc and Chaos 8 3: 619-626 Murashige S and Aihara K (1998b) Experimental study on chaotic motion of a flooded ship in waves. Proc of the Royal Soc of Lond A 454: 2537-2553 Nayfeh AH and Khdeir AA (1986) Nonlinear rolling of ships in regular beam seas. Int Shipbuilding Prog 33:379 Nayfeh AH and Sanchez NE (1990) Stability and complicated rolling responses of ships in regular beam seas. Int Shipbuilding Prog 37 412: 331-352 O’Reilly PJF (1987) Aircraft/deck interface dynamics for destroyers. Marine Technol 24 1: 1525 Papoulias FA (1987) Dynamic analysis of mooring systems. Ph.D. thesis, Dep of Nav Archit and Marine Eng, Univ of Michigan, Ann Arbor, MI

Study of Capsize in Beam Seas 447 Senjanović I, Ciprić G and Parunov J (2000) Survival analysis of fishing vessels rolling in rough seas. Philos Transactions of the Royal Soc of Lond A 358:1943-1965 Soliman MS and Thompson JMT (1991) Transient and steady state analysis of capsize phenomena. Appl Ocean Res 13 2 Spyrou KJ (1996) Homoclinic connections and period doublings of a ship advancing in quartering waves. Chaos 6 2 Story WR (2009) Application of Lyapunov Exponents to Strange Attractors and Intact & Damaged Ship Stability. MS Thesis. Dep of Aerosp and Ocean Eng Virginia Tech Blacksburg VA Wolf A, Swift J, Swinney H and Vastano J (1985) Determining Lyapunov exponents from a time series. Phys D 16: 285-317

Nonlinear Dynamics on Parametric Rolling of Ships in Head Seas Marcelo A.S. Neves*, Jerver E.M. Vivanco*, Claudio A. Rodríguez** *LabOceano, COPPE/UFRJ , **COPPE/UFRJ

Abstract The present paper employs nonlinear dynamics tools in order to investigate the dynamical characteristics governing the complex coupling of the heave, roll and pitch modes in head seas at some regions of the numerical stability map of a fishing vessel. Bifurcation diagrams and Poincaré mappings are computed and employed to investigate the appearance of multistability and chaos associated with increased values of the selected control parameter, the wave amplitude. The connection between these nonlinear characteristics and the coupled nature of the mathematical model are analyzed. Lyapunov exponents corresponding to the three coupled models are computed.

1 Introduction It is well known that parametric rolling in head seas may lead to large roll angles and accelerations in few cycles. Even though many studies on the subject simplify the analysis to a single degree of freedom, there is nowadays a wide acceptance of the relevance of the nonlinear coupling of the roll mode with heave and pitch. In previous studies Neves and Rodríguez (2005, 2006) have introduced a mathematical model in which the heave, roll and pitch motions are nonlinearly coupled to each other. Using this model they investigated the occurrence of head seas parametric rolling on a small fishing vessel. They showed, by means of numerical simulations, comparable to experimental results, the occurrence of strong dependence of the roll responses in head seas conditions on initial conditions, Neves and Rodríguez (2007). In order to investigate the quantitative and qualitative changes of parametric rolling with respect to the encounter frequency tuning and wave amplitude, Neves and Rodríguez (2007a,b) proposed the computation of analytical and numerical

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_25, © Springer Science+Business Media B.V. 2011

449

450 M.A.S. Neves et al.

maps representing the boundaries of stability. The numerical maps aggregate information not only on the boundaries of stability, but also on the amplitude of roll response in the whole region of parametric amplification. The present paper investigates the dynamical characteristics governing the complex coupling of modes at some regions of the numerical stability map. Bifurcation diagrams and Poincaré mappings, Guckenheimer and Holmes (1983), Seydel (1988) are employed in order to investigate the appearance of multistability and chaos associated with increased values of the control parameter wave amplitude. The connection between these nonlinear characteristics and the coupled nature of the model are analyzed. Finally, Lyapunov exponents corresponding to the three coupled models are computed.

2 Mathematical Model Employing Taylor series expansions up to third order, Neves and Rodríguez (2005, 2006) expressed restoring actions in the heave, roll and pitch modes in a coupled way. Wave actions are taken into consideration not only in the Froude-Krilov plus diffraction first order forcing functions, but also in second and third order terms resulting from volumetric changes of the submerged hull due to vertical motions and wave passage effects. The model corresponds to an extension, both in the order of non-linearities and in the levels of coupling, of the model introduced by Paulling and Rosenberg (1959) and Paulling (1961). The equations are taken here in the explicit form described in detail in Neves and Rodríguez (2005, 2006). Thus, the non-linear heave, roll and pitch equations are introduced as:

m Zz z Zz z  Z Z  Zz z  Z  1 Zzz z2  1 Z2  2

2 1 1 1 1 1 Z 2  Zz z  Zzzzz3  Zzz z2  Zz 2 z  Z 2  2 6 2 2 2 1 1 Zz 2 z  Z 3  Zz t z  Z t   Zz t z  Zzz t z2  2 6 Zz t z  Z t  2  Z (t) 2  Zw t 

J

xx

(1)



 K    K   K      K   K z z  K   

1 1 1 K zz z 2  K  3  K  2  K z z  K  t   2 6 2 K  t   K z t z  K  t   0

(2)

Nonlinear Dynamics on Parametric Rolling of Ships in Head Seas 451

J

yy

 M    M   M zz  M z z  M z z  M  

1 1 1 1 M zz z 2  M  2  M  2  M z z  M zzz z 3  2 2 2 6 1 1 1 1 M zz z 2  M z 2 z  M  2  M z 2 z  2 2 2 2 1 M  3  M z t z  M  t   M z t z  6 M zz t z 2  M  t   M z t z  M  t  2 

(3)

M  (t ) 2  M w t  On the left hand side of Eqs. (1-3), added masses and wave damping terms are assumed linear. A quadratic roll damping is considered in equation (2). The terms associated with variables z ,  ,  and wave elevation  (t ) correspond to the non-zero linear and non-linear (up to third order) coefficients due to hydrostatic and wave pressure effects analytically derived by Neves and Rodríguez (2005, 2006), dependent on hull characteristics and on wave amplitude, frequency and T time. On the right hand side of Eqs. (1-3), Z W (t ) K W (t ) M W (t ) represent linear wave excitation. Due to the particular wave incidence considered,

KW  0

has been assumed in Eq. (2). Once all the coefficients in Eqs. (1-3) are known, this set of three equations may be numerically integrated without difficulty.

3 Numerical Limits of Stability Numerical simulations have been performed in the case of a fishing vessel denominated TS. Details of the ship are given in Fig. 1 and Table 1. Parametric rolling of this fishing vessel has been experimentally and numerically examined in detail in Neves and Rodríguez (2005, 2006), Neves et al. (2002). Numerical simulations performed using equations (1-3) have been successfully compared to experimental results for different wave conditions and ship speeds. It was verified that the fishing vessel employed in the present investigation is quite prone to strong parametric rolling in head seas.

452 M.A.S. Neves et al.

Fig. 1 Hull form of transom stern (TS) fishing vessel

Table 1 Ship main characteristics

Denomination

Ship TS

Overall length

[m]

25.91

Length between perpendiculars

[m]

22.09

Breadth

[m]

6.86

Depth

[m]

3.35

Draft

[m]

2.48

Displacement

[ton] 170.3

Longitudinal radius of gyration

[m]

5.52

Metacentric height

[m]

0.37

Table 2 Two sets of initial conditions

z0 ( m ) z0 (m / sec) 0 (deg)  (deg/ sec) 0

 0 (deg) 0 (deg/ sec)

I.C.#01 0.00

I.C.#02 0.00

0.01

0.01

2.00

2.50

-0.50

-0.80

0.00

0.00

0.01

0.01

Nonlinear Dynamics on Parametric Rolling of Ships in Head Seas 453

It is important to investigate parametric rolling not only at the exact encounter frequency tuning we / wn 4  2.0 . In fact, large amplifications may take place in a quite broad spectrum of excitation frequencies. In order to comprehensively investigate the unstable regions Neves and Rodríguez (2007b) proposed the computation of numerical maps representing the boundaries of stability but containing information on the amplitude of roll response in the whole region of parametric amplification. Figure 3 shows the limits of stability of the fishing vessel in head seas at Fn = 0.30, corresponding to the first region of instability. The mapping is constructed by numerically computing the roll amplitude for different encounter frequencies and wave amplitudes. All points of the map are computed for the set of initial conditions I.C.#01 defined in Table 2. The intensity of the final roll amplitude is indicated by the color scale displayed on the right hand side of the figure. Four important features of the new limits of stability are:  The appearance of upper boundaries, indicating that for increased wave amplitudes, parametric rolling may not necessarily increase; in fact, it tends to disappear.  A general tendency of the unstable area to bend to the right, indicating that the exact tuning we / wn 4  2.0 is not necessarily the tuning with stronger amplifications.  Smooth growth of roll amplitude at lower level of boundaries, abysmal (sudden) decrease in the upper boundaries.  Upper boundaries with fractal geometry.  Larger area of instability as the roll initial conditions were modified.

Fig. 2 Limits of stability, ship TS, Fn=0.30, I.C.#01

454 M.A.S. Neves et al.

4 Bifurcation Diagrams In Figure 2 the whole spectrum of exciting frequencies was explored. It has been observed that distinct characteristics are revealed. It is now desirable to get an in depth knowledge of dynamical characteristics as the parameter wave amplitude AW is increased. So, if previously we have been more interested in the limits of stability as a whole, now we wish to have a closer look at some domains inside the unstable area. For this purpose, we will investigate the changes in dynamic characteristics as we cross the area inside the limits. A limited region of the map of limits of stability will be explored, that is, we will follow a vertical line defined at the tuning we / wn 4  2.0 in Figure 2. Aiming at demonstrating the influence of initial conditions on the solutions corresponding to points inside the area of the limits of stability we developed a brute-force algorithm for capturing the branching of solutions for a specified set of initial conditions. Thus, using the AW parameter, the type of bifurcation diagrams as shown in Figure 3 for the heave, roll and pitch motions are developed for the same set of initial conditions used in the mapping of the limits of stability. These diagrams reveal the branching structure for distinct ranges of wave amplitude; qualitative and quantitative types of responses are noticed. These distinct results may be summarized as shown in Table 3.

Fig. 3 Heave, roll and pitch bifurcation diagrams, Fn=0.30, I.C.#01

Nonlinear Dynamics on Parametric Rolling of Ships in Head Seas 455 Table 3 Roll solutions for Fn=0.30

Range of Aw (m) 0.0000 - 0.6036 0.6037 - 0.6129 0.6130 - 0.6626 0.6627 - 0.6758 0.6759 - 0.6782 0.6783 - 0.67881 0.67882 - 0.7000

Type of roll response Typically linear period-3 Multistability, period-1 Multistability, period-2 Multistability, period-4 Multistability, period-8 Chaos

Two interesting characteristics, not observable in the numerical limits of stability, are revealed by this bifurcation analysis. First, in the short second range of AW one observes the appearance of a solution with 3 periods that ends with a sudden appearance of a burst of non-periodic solutions. The period-3 solutions of heave, roll and pitch motions are illustrated in Figures 4-6, respectively. In each of them time history, phase plane and Poincaré map are shown. The appearance of non-periodic solutions is illustrated in Figure 7 which shows the roll time series, the corresponding phase diagram and Poincaré map for Aw=0.6129 m.

Fig. 4 Heave motion, phase plane and Poincaré map, period-3 solution, Aw=0.605 m, I.C.#01

Fig. 5 Roll (max=24.18°), phase plane and Poincaré map, period-3 solution, Aw=0.61 m, I.C.#01

456 M.A.S. Neves et al.

Fig. 6 Pitch motion, phase plane and Poincaré map, period-3 solution, Aw=0.605 m, I.C.#01

Fig. 7 Roll motion, phase plane and Poincaré map, Aw=0.6129 m, I.C.#01. Non-periodic solutions

The second interesting characteristic encountered is multistability with associated alternance of values. This dynamical feature arises immediately after the occurrence of a burst of non-periodic solutions, as shown in Figure 3. In the bifurcation diagram one may get the impression that the motion has migrated to a period-2 solution, but a detailed analysis will show that this is not the case. In fact the roll solutions in the third range of Aw are period-1, but as illustrated in Figures 8 and 9, the solutions continuously alternate from one attractor to another one which is situated close by, at each new value of the parameter Aw. In other words, roll motion either lives in one attractor or in the other, but always with a single period. Subsequently, for higher wave amplitudes, flip bifurcation will take place together with multistability: period-2, 4 and 8 solutions will appear in sequence ending in chaos.

Fig. 8 Roll motion: (a) Aw=0.639 m, (b) Aw=0.6391 m. Multistability for two neighboring points, I.C.#01

Nonlinear Dynamics on Parametric Rolling of Ships in Head Seas 457

It is interesting to observe that in this third range of Aw (0.6130 - 0.6626) the roll motion undergoes multistability with period-1 solutions, as shown in Figures 8 and 9. But in this same range, the vertical motions have already undergone a period doubling bifurcation. This is shown in Figures 10 and 11 for heave and pitch, respectively. Another aspect worth noting is that the alternating process does not contaminate these modes, Vivanco (2009). Subsequently, in the fourth range of Aw (0.6627 - 0.6758) the roll motion continues with multistability but responding with period-2 solutions, as shown in Figure 12, whereas the heave and pitch motions now respond with period-4 solutions. The sequence of flip bifurcations soon leads the coupled system to respond with chaotic motions. Figure 13 illustrates the period-4 roll motion and finally, Figure 14 shows the chaotic behaviour for Aw=0.683 m. The region with chaotic behaviour ends abruptly at the wave amplitude corresponding to the upper limit of stability of Figure 2.

Fig. 9 Roll phase planes: (a) Aw=0.639 m, (b) Aw=0.6391 m. Multistability for two neighboring points, I.C.#01

Fig. 10 Heave time history, phase plane and Poincaré map, period-2 solution, Aw=0.64 m, I.C.#01

458 M.A.S. Neves et al.

Fig. 11 Pitch time history, phase plane and Poincaré map, period-2 solution, Aw=0.64 m, I.C.#01

Fig. 12 Roll motion (max=24.27°), phase plane and Poincaré map, period-2 solution for Aw=0.67 m, I.C.#01

Fig. 13 Roll motion ( 

max=24.73°),

phase plane and Poincaré map, period-4 solution for

Aw=0.678 m, I.C.#01

Fig. 14 Roll motion ( 

max=25°),

phase diagram and Poincaré map, chaotic behaviour for Aw=0.683 m, I.C.#01

Nonlinear Dynamics on Parametric Rolling of Ships in Head Seas 459

Fig. 15 Roll bifurcation diagram, Fn=0.30, I.C.#02

To illustrate the dependency of the limits of stability on initial conditions, a second set of initial conditions (I.C.#02) defined in Table 2 is considered. When initial conditions are changed, the general bifurcation set is preserved, but it may be observed in the new bifurcation diagram that, as shown in Figure 15, the range of responses with period-3 solution simply is not reached. At AW  0.617 m the motion jumps to a higher value. Instead of alternating continuously, the roll motion now undergoes period-1 motions with jumps, the first one producing a jump from 17 to 21 degrees. A second jump takes place at AW  0.648m , from 24 to 16 degrees. At AW  0.663m the motion begins a flip bifurcation at the same point as obtained in Figure 3. Period-2, 4, 8 solutions appear in cascade, at all stages featuring multistability. Finally, chaotic motions take place again for AW  0.6788m .

5 Lyapunov Exponents Lyapunov exponents offer a quantitative measure of the sensitivity of a nonlinear dynamic system to initial conditions. As such, they characterize the chaotic behaviour of the system. The Lyapunov exponents provide information on the global stability of the system: values obtained for these exponents, after the transients, are negative for stable zones, zero at bifurcation and positive in the chaotic zones. The method of trajectories developed by Wolf and Vastago (1985) is employed here in order to compute all the exponents relative to the heave, roll and pitch coupled motions for the tuning we / wn 4  2.0 and different wave

460 M.A.S. Neves et al.

amplitudes. Figures 16 and 17 show the six exponents for two wave amplitudes, Aw=0.62 m and Aw=0.63 m, respectively. In Figure 16 all exponents are negative, with the larger one being quite close to zero, whereas in Figure 17 this larger exponents is practically zero. Compiling the values of the Lyapunov exponents obtained for varying wave amplitudes at t=800 sec, Figure 18 gives a complete picture of the evolution of the first three larger exponents, with the control parameter Aw, showing that at Aw=0.63 m the system undergoes chaos. This indicates that other attractors may be competing with those observed in Figures 3 and 15. In any case, this result demonstrates that the motions shown in Figure 7, corresponding to Aw=0.6129 m are non-periodic, not chaotic.

Fig. 16 Lyapunov exponents, Aw=0.62 m

6 Conclusions Numerical limits of stability for a fishing vessel at Fn=0.30 undergoing strong parametric rolling in head seas have been computed for a range of encounter frequencies. The main dynamical characteristics of these limits have been discussed.

Fig. 17 Lyapunov exponents, Aw=0.63 m

Nonlinear Dynamics on Parametric Rolling of Ships in Head Seas 461

Fig. 18 Lyapunov exponents (t=800 sec) for different wave amplitudes

For the encounter frequency tuning corresponding to the first region of instability of the Mathieu stability map, bifurcation diagrams for the heave, roll and pitch motions have been computed considering wave amplitude as control parameter. Interesting phenomena such as coexistence of attractors with period-3 solutions, appearance of a burst of non-periodic solutions, multistability with alternance, fold and flip bifurcation and chaos have been identified. The phase planes and Poincaré mappings showed that the period-3 solutions and burst of non-periodic solutions are common to the three modes of motion considered. On the other hand, multistability with alternance only takes place for the roll motion. It was observed that when a different set of initial conditions was considered, both the non-periodic motions and alternance disappeared. Finally, Lyapunov exponents have been computed for the same encounter frequency tuning, again taking the wave amplitude as control parameter.

Acknowledgements The present investigation is supported by CNPq within the STAB project (Nonlinear Stability of Ships). The Authors also acknowledge financial support from CAPES, FAPERJ and LabOceano. Thanks are due to Prof. Marcelo A. Savi for many fruitful discussions.

462 M.A.S. Neves et al.

References Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Appl Math Sci 42 Springer-Verlag Neves M A S, Rodríguez C (2005) A nonlinear mathematical model of higher order for strong parametric resonance of the roll motion of ships in waves. Mar Syst & Ocean Tech – J Sobena 1 2:69-81 Neves M A S, Rodríguez C (2006) Unstable ship motions resulting from strong nonlinear coupling. Ocean Eng 33:99-108 Neves M A S, Rodríguez C (2007a) Influence of nonlinearities on the limits of stability of ships rolling in head seas. Ocean Eng 34:1618-1630 Neves M A S, Rodríguez C (2007b) An investigation on roll parametric resonance in regular waves. Int Shipbuild Prog 54:207-225 Neves M A S, Pérez N A, Lorca O (2002) Experimental analysis on parametric resonance for two fishing vessels in head seas. 6th Int Ship Stab Workshop Webb Inst NY Paulling J R, Rosenberg R M (1959) On unstable ship motions resulting from nonlinear coupling. J of Sh Res 3 1:36-46 Paulling J R (1961) The transverse stability of a ship in a longitudinal seaway. J of Ship Res 4 4Mar:37-49 Seydel R (1988) From equilibrium to chaos: practical bifurcation and stability analysis. Elsevier Sci Publ NY Vivanco J E M (2009) Parametric rolling of a fishing vessel - nonlinear dynamics. M.Sc. Diss COPPE/UFRJ Jan (Port) Wolf A, Vastago A (1985) Determining Llyapunov exponents from a time series. Phys Rev 16:25-317

6 Roll Damping

A Simple Prediction Formula of Roll Damping of Conventional Cargo Ships on the Basis of Ikeda’s Method and Its Limitation Yuki Kawahara, Kazuya Maekawa, Yoshiho Ikeda Osaka Prefecture University, Sakai, Japan

Abstract Since the roll damping of ships has significant effects of viscosity, it is difficult to calculate it theoretically. Therefore experimental results or some prediction methods are used to get the roll damping in design stage of ships. Among some prediction methods, Ikeda’s one is widely used in many ship motion computer programs. Using the method, the roll damping of various ship hulls with various bilge keels are calculated to investigate its characteristics. Ship hull forms are systematically changed by changing length, beam, draft, midship sectional coefficient and prismatic coefficient. On the basis of these predicted roll damping of various ships, a very simple prediction formula of the roll damping of ships is deduced. It is found, however, that this formula cannot be used for ships that have higher position of the center of gravity.

1 Introduction In 1970’s, strip methods for predicting ship motions in 5-degree of freedoms in waves have been established. The methods are based on potential flow theories (Ursell-Tasai method, source distribution method and etc), and can predict pitch, heave, sway and yaw motions of ships in waves in fairly good accuracy. In roll motion, however, the strip methods do not work well because of significant viscous effects on the roll damping. Therefore, some empirical formulas or experimental data are used to predict the roll damping in the strip methods. To improve the prediction of roll motions by these strip methods, one of the authors carried out a research project to develop a roll damping prediction method which has the same concept and the same order of accuracy as the strip methods which are based on hydrodynamic forces acting on strips, or cross sections of a ship [Ikeda et al. (1976), Ikeda et al. (1977a), Ikeda et al. (1977b), Ikeda et al. (1978): All papers in English versions were published as Reports of Dept. of Naval M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_26, © Springer Science+Business Media B.V. 2011

465

466 Y. Kawahara et al.

Architecture, Univ. of Osaka Pref.]. The review of the prediction method was made by Himeno (1981) and Ikeda (1982) with the computer program. The prediction method, which is now called Ikeda’s method, divides the roll damping into the frictional (BF), the wave (BW), the eddy (BE) and the bilge keel (BBK) components at zero forward speed, and at forward speed, the lift (BL) is added. Increases of wave and friction components due to advance speed are also corrected on the basis of experimental results. Then the roll damping coefficient B44 (=Roll damping moment (kgfm)/Roll angular velocity (rad/sec)) can be expressed as follows.

B44  B F  BW  B E  B L  B BK

(1)

At zero forward speed, each component except the friction component is predicted for each cross section with unit length and the predicted values are summed up along the ship length. The friction component is predicted by Kato’s formula for a three-dimensional ship shape. Modification functions for predicting the forward speed effects on the roll damping components are developed for the friction, wave and eddy components. The computer program of the method was published, and the method has been widely used. For these thirty years, the original Ikeda’s method developed for conventional cargo ships has been improved to apply many kinds of ships, for examples, more slender and round ships, fishing boats, barges, ships with skegs and so on. The original method is also widely used. However, sometimes, different conclusions of roll motions were derived even though the same Ikeda’s method was used in the calculations. Then, to check the accuracy of the computer programs of the same Ikeda’s method, a more simple prediction method with the almost same accuracy as the Ikeda’s original one has been expected to be developed. It is said that in design stages of ships, Ikeda’s method is too complicated to use. To meet these needs, a simple roll damping prediction method was deduced by using regression analysis [Kawahara et al. (2008)].

2 Previous Prediction Formula The simple prediction formula proposed in the previous paper [Kawahara (2008)] cannot be used for modern ships that have high position of center of gravity or long natural roll period such as large passenger ships with relatively flat hull shape. In order to investigate its limitation, the authors compared with the result of this prediction method and original Ikeda’s one out of its calculating limitation. Figure 1 shows the result of the comparison with these method of the roll damping. The upper one is on the condition that the center of gravity is low and the lower one on the condition that the center of gravity is high.

A Simple Prediction Formula of Roll Damping of Conventional Cargo Ships

467

(a) OG/d = -0.2

(b) OG/d = -1.5 Fig. 1 Comparison between Ikeda’s method and proposed one of roll damping at L/B=6.0 B/d=4.0, Cb=0.65, Cm=0.98, φa=10°, bBK/B=0.025 and lBK/Lpp=0.2. (OG denotes the distance between water surface and center of gravity, and defined plus when the center of gravity is below water surface.)

From this figure, the roll damping estimated by this prediction formula are in good agreement with the roll damping calculated by the Ikeda’s method for low position of center of gravity, but the error margin grows for the high position of center of gravity. The results suggest that the previous prediction formula is necessary to be revised.

3 Methodical Series Ships Modified prediction formula will be developed on the basis of the predicted results by Ikeda’s method using the methodical series ships. This series ships are constructed based on the Taylor Standard Series and its hull shapes are methodically changed by changing length, beam, draft, midship sectional coefficient and longitudinal prismatic coefficient. The geometries of the series ships are given by the following equations.

468 Y. Kawahara et al.

y1  Q( x)  C P P ( x)  f11tT ( x)  nN ( x)  f12 F ( x)

(2)

y2  Q( x)  CW P( x)  f 21tT ( x)  nN ( x)  f 22 F ( x)

(3)

where

Q ( x )  30 x 2  100 x 3  105 x 4  36 x 5

(4)

P ( x )  60 x 2  180 x 3  180 x 4  60 x 5

(5)

T ( x )  x  6 x 2  12 x 3  10 x 4  3 x 5

(6)

N ( x)  0.5 x 2  2 x 3  2.5 x 4  x 5

(7)

F ( x)  26.562 x 6  105.74 x 5  162.71x 4  116.58 x 3  34.532 x 2  0.6998 x  0.7923 t  3969.7C P6  16664.6C P5  28230C P4  24951C P3  12205C P2  3147.7C P  335.67

(8) (9)

(at stern side and CP<0.73)

t  113.64C P2  149.68C P  50.221

(10)

t  96.339C P5  173.59C P4  159.75C P3

(11)

(at stern side and CP ≥0.73)

 113.4C P2  54.123C P  10.686

(at bow side and CP <0.72)

t  41.667C P2  49.167C P  14.9

(12)

n  5.7035C P3  30.16C P2  33.471C P  10.606

(13)

n  10.417C P2  13.458C P  4.305

(14)

n  1664.6C P5  5596.7C P4  7413.2C P3

(15)

(at bow side and CP ≥0.72)

(at stern side and CP <0.74) (at stern side and CP ≥0.74)

 4815.6C P2  1525.3C P  186.79

(at bow side and CP <0.73)

n  0.625C P  0.4312 (at bow side and CP ≥0.73)

(16)

A Simple Prediction Formula of Roll Damping of Conventional Cargo Ships

469

where y1 is the non-dimensional value obtained by sectional-immersed-area divided by maximum-immersed-area, y2 is the non-dimensional value obtained by sectional-water-line-breadth divided by maximum-water-line-breadth, x is the non-dimensional value of longitudinal position when x is measured from the extremity of either the bow or stern, CP is the longitudinal prismatic coefficient, CW is the water-plane coefficient, f11 is constant and equal to 0.6 for the stern and 1.0 for the bow, f12 is constant and equal to 0.05, f21 is constant and equal to 2.0 for the stern and 1.0 for the bow, f22 is constant and equal to 0.15 for the stern and 0.1 for the bow. However, occasionally, corrections of the results obtained from the Eqs. 2, 3 are required. For example, if maximum of y1 exceeds 1.0, it is adjusted to 1.0. Figures 2 and 3 show the sectional area curves obtained from Eq. 2 and a body plan of a ship with Cb=0.84 (Cb : block coefficient) and Cm=0.98 (Cm : midship section coefficient), respectively. Since the hull shapes used here are conventional ones as shown in Figure 3, an application of the deduced prediction method to modern unconventional hull shapes, for example buttock-flow hull, should be careful.

Fig. 2 Sectional-area curves of series model used in calculation

Fig. 3 An example of body plan of ship with Cb=0.84 and Cm=0.98

470 Y. Kawahara et al.

4 Proposal of New Prediction Method of Roll Damping In this chapter, the characteristics of each component of the roll damping, the frictional, the wave, the eddy and the bilge keel components at zero advanced speed, are discussed, and a simple prediction formula of each component is developed. The roll damping coefficient (B44) and circular frequency (ω=2π/Tw) are defined as follows,

B Bˆ44  44 2 ∇ B

ˆ  

B 2g

(17)

B 2g

(18)

where ρ denotes water density, ∇ displacement volume, B beam and g is gravity acceleration, respectively. The relationship between B44 and N coefficient (Bertin) is as follows. Bˆ N  Bˆ 44 ( a in Eq. 19 is in deg .) GM a

(19)

4.1 Frictional Component (BF) In Ikeda’s method, the friction damping at Fn=0 is given by Kato’s formula as follows,

BF 

4 s f r f3 a c f 3

(20)

where cf is frictional coefficient, rf is average radius from the axis of rolling and sf is wetted surface area. These parameters in the equation are given by the following equations, 1

 3.22r f2 a2  2  c f  1.328   T   0.887  0.145Cb 1.7d  Cb B   2OG rf 



s f  LPP 1.75d  Cb B 

(21)

(22) (23)

A Simple Prediction Formula of Roll Damping of Conventional Cargo Ships

471

where φa denotes roll amplitude, T roll period, ν dynamic coefficient of viscosity, OG distance from calm water surface to the axis of rolling (downward direction is positive) and d draft, respectively. In the present study, since the frictional component of roll damping is already given for a whole ship as Eqs. 20-23, the formula is used without any modification in the simple prediction method developed in the present study. It should be noted, however, the frictional component is negligible for a full-scale ship although it takes about 5-10% of the total roll damping for a small scale model.

4.2 Wave Component (Bw) As well known, the wave component of the roll damping for a two-dimensional cross section can be calculated by potential flow theories in fairly good accuracy. In Ikeda’s method, the wave damping of a strip section is not calculated and the calculated values by any potential flow theories are used as the wave damping. 4.2.1 Characteristics of the Wave Component The reason why viscous effects are significant in only roll damping can be explained as follows. Figure 4 shows the wave component of the roll damping for 2-D sections calculated by a potential flow theory. We can see that the component is very small for sections with half breadth/draft ratio, H0=05-1.5, particularly when area coefficient, σ is large. Usually, conventional ships have such H0 at parallel middle body part. Then ships have relatively small wave roll damping, and viscous effects play an important role in the roll damping. In Figures 5 and 6, calculated distributions of the wave component of the roll damping are shown. The results for a full ship (Cb=0.8) shown in Figure 5 demonstrate that the damping created by the mid-ship body is very small, and that created by the stern body is large. The results for a slender ship (Cb=0.5) shows two peaks at the cross sections of SS=3 and SS=7. The reason why the peaks appear on bow and stern parts can be easily understood from the calculated results shown in Figure 4. Dependencies of the wave component on location of roll axis or center of gravity are shown in Figure 7.

472 Y. Kawahara et al.

Fig. 4 Characteristics of roll damping coefficient of wave component for two-dimensional section at OG/d=0 and ˆ =1.25

Fig. 5 Longitudinal distribution of σ, H0 and roll damping coefficient of wave component for a ship with L/B=5, B/d=3.5, Cm=0.94, Cb=0.8,OG/d=0 and ˆ =1.25

A Simple Prediction Formula of Roll Damping of Conventional Cargo Ships

473

Fig. 6 Longitudinal distribution of σ, H0 and roll damping coefficient of wave component for a ship with L/B=5, B/d=3.5, Cm=0.94, Cb=0.5, OG/d=0 and ˆ =1.25

Fig. 7 Characteristics of roll damping coefficient of wave component for a whole ship at L/B=5, B/d=3.5, Cm=0.94, Cb=0.8

4.2.2 Proposed Formula of the Wave Roll Damping By fitting these predicted wave components a simple prediction formula is deduced as follows.

474 Y. Kawahara et al. x1  B d , x2  Cb , x3  Cm , x4  1  OG d , x5  ωˆ BˆW  A1 x5  exp (-A2(LOG(x5 )-A3 )2 / 1.44 ) 2

A1=(A11 x 4 +A12 x4+A13 )AA1 3

2

A2=-1.402 x4 +7.189 x4 -10.993 x4+9.45 6

5

4

3

2

A3=A31 x4 +A32 x4 +A33 x4 +A34 x4 +A35 x4 +A36 x 4+A37+AA3 x6=x4-AA32 AA1  ( AA11 x3  AA12 )  (1  x4 )  1.0 9

8

7

AA3=AA31(-1.05584 x6 +12.688 x6 -63.70534 x6 + 6

5

4

172.84571x6 -274.05701x6 +257.68705 x6 -141.40915 x6 2

3

2

+44.13177 x6 -7.1654 x6-0.0495 x1 +0.4518 x1-0.61655 ) 3

2

AA31=(-0.3767 x1 +3.39 x1 -10.356 x1+11.588 )  AA311 2

AA32=-0.0727 x1 +0.7 x1-1.2818 3

2

AA311=(-17.102 x2 +41.495 x2 -33.234 x2+8.8007 )  x4+ 3

2

36.566 x2 -89.203 x2 +71.8 x2-18.108 6

5

4

A31=-7686.0287 x2 +30131.5678 x2 -49048.9664 x2  3

2

42480.7709 x2 -20665.147 x2 +5355.2035 x2-577.8827 6

5

4

A32=61639.9103 x2 -241201.0598 x2 +392579.5937 x2  3

2

340629.4699 x2 +166348.6917 x2 -43358.7938 x2+4714.7918 6

5

A33=-130677 .4903 x2 +507996.2604 x2 -826728.7127 x2 3

 722677.104 x2 -358360.7392 x2

4

2

+95501.4948 x2-10682.8619 6

5

A34=-110034.6584 x2 +446051.22 x2 -724186.4643 x2 3

 599411.9264 x2 -264294.7189 x2

4

2

+58039.7328 x2-4774.6414 6

5

A35=709672.0656 x2  2803850.2395 x2 +4553780.5017 x2 3

 3888378.9905 x2 +1839829.259 x2

2

 457313.6939 x2+46600.823 6

5

A36=-822735.9289 x2 +3238899.7308 x2 -5256636.5472 x2 3

 4500543.147 x2  2143487 .3508 x2 +538548.1194 x2  55751.1528

2

4

4

A Simple Prediction Formula of Roll Damping of Conventional Cargo Ships 6

5

A37=299122.8727 x 2 -1175773.1606 x 2 +1907356.1357 x 2 3

 1634256.8172 x 2 +780020.9393 x 2

475

4

2

 196679.7143 x 2+20467 .0904 2

AA11=AA111 x 2 +AA112 x 2+AA113 2

AA12=AA121 x 2 +AA122 x 2+AA123 3

2

AA111=17.945 x1 -166.294 x1 +489.799 x1-493.142 3

2

AA112=- 25.507 x1 +236.275 x1 -698.683 x1+701.494 3

2

AA113=9.077 x1 -84.332 x1 +249.983 x1- 250.787 3

2

AA121=-16.872 x1 +156.399 x1 - 460.689 x1+463.848 3

2

AA122=24.015 x1 -222.507 x1 +658.027 x1-660.665 3

2

AA123=-8.56 x1 +79.549 x1 - 235.827 x1+236.579 2

A11=A111 x 2 +A112 x 2+A113 3

2

A12=A121 x 2 +A122 x 2 +A123 x 2+A124 3

2

A13=A131 x 2 +A132 x 2 +A133 x 2+A134 3

2

A111=-0.002222 x1 +0.040871x1 -0.286866 x1+0.599424 3

2

A112=0.010185 x1 -0.161176 x1 +0.904989 x1-1.641389 3

2

A113=-0.015422 x1 +0.220371x1 -1.084987 x1+1.834167 4

3

A121=-0.0628667 x1 +0.4989259 x1 +0.52735 x1

2

-10.7918672 x1+16.616327 4

3

A122=0.1140667 x1 -0.8108963 x1 -2.2186833 x1

2

+25.1269741x1-37.7729778 4

3

A123=-0.0589333 x1 +0.2639704 x1 +3.1949667 x1

2

-21.8126569 x1+31.4113508 4

3

A124=0.0107667 x1 +0.0018704 x1 -1.2494083 x1

2

+6.9427931x1-10.2018992 3

2

A131=0.192207 x1 - 2.787462 x1 +12.507855 x1-14.764856 3

2

A132=-0.350563 x1 +5.222348 x1 -23.974852 x1+29.007851 3

2

A133=0.237096 x1 -3.535062 x1 +16.368376 x1-20.539908 3

2

A134=-0.067119 x1 +0.966362 x1 -4.407535 x1+5.894703

 0.5  C b  0.85, 2.5  B / d  4.5, ˆ  1.0      1.5  OG / d  0.2, 0.9  C m  0.99 

(24)

476 Y. Kawahara et al.

4.3 Eddy Component (BE) The eddy component of the roll damping is created by small separation bubbles or small shedding vortices generated at the bilge part of midship section and large vortices generated at the relatively sharp bottom of bow and stern sections. Although vortex shedding flow from oscillating bluff bodies is usually govern by Keulegan-Carpenter number, Kc, it was found by Ikeda et al. (1978b) that the viscous forces created by such small separation bubbles or small shedding vortices do not significantly depend on Kc. In Ikeda’s prediction method, the distribution of the pressure created on a hull surface by such separation bubble is assumed as a simple shape for each shape of cross sections on the basis of experimental results of pressure distribution on hull surfaces. The pressure value was determined as the calculated eddy components of the roll damping for various cross sections are in good agreement with measured ones. Then, the eddy damping of a strip section is calculated by following formulas.

 R  OG R  1  f1 d 1  d  f1 d    4  2 LPP d 2rmax a  BE  CP 0 2 3   f  H  f R    2  0 1 d  (25)

1 1  tanh20  0.7  2 1 f 2  1  cos    1.5 1  e 51  sin 2  2 1 C P 0  0.87 e   4e  0.1877  3 2 B H0  2d f1 





 

S Bd





(26) (27) (28) (29) (30)

where, rmax is maximum distance from center of gravity (roll axis) to hull surface, R is bilge radius and σ is sectional area coefficient. The eddy roll damping of a whole ship is calculated by integrating BE’ to longitudinal direction.

A Simple Prediction Formula of Roll Damping of Conventional Cargo Ships

477

4.3.1 Characteristics of Eddy Component In Figures 8 and 9, the longitudinal distribution of the predicted eddy component of the roll damping are shown for a slender ship (Cb=0.5) and a full ship (Cb=0.8). The eddy component of a slender ship is large at bow and stern parts, and has a small peak at mid-ship, as shown in Figure 8. For a full ship, however, the eddy damping at parallel middle body part becomes larger and longer as shown in Figure 9. The component at stern is large as a slender ship, but becomes small at bow part. These may be caused by that both ships has similar V shape stern but the full ship has U shape bow section which is not thin and flat shape as a slender ship. Figure 10 shows the effect of midship-section coefficient (Cm ) on the eddy damping. The results demonstrate that the eddy component of the roll damping of parallel body parts of ships is significantly sensitive to Cm. This is because of larger flow separation occurs at bilge corners of ships with large midship-section coefficient.

Fig. 8 Longitudinal distribution of σ, H0 and roll damping coefficient of eddy component for a ship with L/B=5, B/d=3.5, Cm=0.94, Cb=0.5,OG/d=0 and ˆ =1.25

478 Y. Kawahara et al.

Fig. 9 Longitudinal distribution of σ, H0 and roll damping coefficient of eddy component for a ship with L/B=5, B/d=3.5, Cm=0.94, Cb=0.8, OG/d=0 and ˆ =1.25

Fig. 10 Effects of midship-section coefficient, Cm on eddy component of roll damping for slender ships (Cb=0.5) and full ships (Cb=0.8)

4.3.2 Proposed Formula of the Eddy Roll Damping By fitting these predicted eddy components a simple prediction formula is deduced as follows.

A Simple Prediction Formula of Roll Damping of Conventional Cargo Ships

479

x1  B d , x 2  C b , x 3  C m , x 4  OG d Bˆ E 

4 L pp d 4 ˆ  a 3  B

2

CR 

4 ˆ  a 3  x 2  x1

C R  AE  exp( B E1  B E 2  x 3

BE 3

3

CR

)

AE  (0.0182 x 2  0.0155)  ( x1  1.8) 3  79.414 x 2 3

4

2

 215.695 x 2  215.883x 2  93.894 x 2  14.848 B E1  (0.2 x1  1.6)  (3.98 x 2  5.1525)  x 4



2



 (0.9717 x 2  1.55 x 2  0.723)  x 4  (0.04567 x 2  0.9408) B E 2  (0.25 x 4  0.95)  x 4 3

2

 219.2 x 2  443.7 x 2  283.3x 2  59.6 B E 3  (46.5  15 x1 )  x 2  11.2 x1  28.6  0.5  C b  0.85, 2.5  B / d  4.5     1 . 5  /  0 . 2 , 0 . 9   0 . 99 OG d C m  

(31)

4.4 Bilge Keel Component (BBK) The bilge keel component is usually the largest one in the roll damping. The component creates 50-80% of the total roll damping. The component is created by shedding vortices from the sharp edges of bilge keels due to roll motion. The component can be divided into two components, the normal force component (BN) and the hull pressure component (BS). Both components are created by the same vortices from the edge of bilge keels. The former one is created by the force acting a bilge keel, and the latter by the pressure over the hull surfaces in front and back sides of the bilge keel. In Ikeda’s method, the pressure distributions in front and back of a bilge keel are assumed on the basis of the measured ones, and are integrated over the hull surface. This means that the method may be available for any shape of cross section. Ikeda et al. (1976) experimentally found that the magnitude and distribution of the pressure created by a bilge keel significantly depends on Kc. The roll damping due to the normal force acting on bilge keel is given by following expressions.

480 Y. Kawahara et al.

BN 

8 r 3l BK bBK  af 3

2

  b  22.5 BK  2.4  rf a  

f  1  0.3 exp 1601   

(32) (33)

where r is distance from the axis of rolling to bilge keel, bBK is width of bilge keel, lBK is length of bilge keel and f is a correction factor on bilge radius to take the velocity increase there into account. The roll damping due to the hull pressure created by bilge keels is calculated by following equations. BS 

I

4 r 2 d 2  a f 2 I 3

1 C P1l BK ds d2 

(34) (35)

where CP1 is pressure coefficient on hull surface. The positive pressure coefficient (CP1+) of face of a bilge keel and the negative pressure coefficient (CP1-) of back side of a bilge keel are given by following formulas.

C P1  1.2 C P1  

22.5bBK  1.2 rf a

(36) (37)

4.4.1 Characteristics of the Bilge Keel Component Calculated bilge keel components of the roll damping by Ikeda’s method are shown in Figs. 11-13 to demonstrate the characteristics. In Figure 11 the bilge keel component of the roll damping of slender ships (Cb=0.58) and full ships (Cb=0.81) with different midship-section coefficients are shown. In the prediction the area of bilge keels is systematically changed. The results show that the roll damping component increases with increasing area of bilge keels. For full ships the increase of the component is almost linear, but for slender ships it shows non-linearly increases. The magnitude of the component significantly depends on midship-section coefficients, but not so sensitive on block coefficient Cb.

A Simple Prediction Formula of Roll Damping of Conventional Cargo Ships

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Fig. 11 Bilge keel component of roll damping of slender and full ships for various bilge keel lengths and constant breadth (bBK)

Fig. 12 Effect of location of center of gravity, or roll axis on bilge keel component of roll damping.

Fig. 13 Bilge keel component of roll damping of slender and full ships with bilge keel of the same area (SBK) but different lengths.

In Figure 12, effects of location of center of gravity on the bilge keel component of the roll damping are shown for slender and full ships. As height of center of gravity decreases, or OG/d increases, the bilge keel component decreases. This is because

482 Y. Kawahara et al.

relative flow speed at bilge corner decreases and flow separation at the edge of bilge keels weakens. In Figure 13, effects of aspect ratio (=length/breadth of a bilge keel) on the roll damping of ships with the same area bilge keels are shown. It can be seen that the component is not so sensitive with aspect ratio, but increases with increasing aspect ratio, or more slender bilge keels for full ships with large midship-section coefficients. 4.4.2 Proposed Formula of the Bilge Keel Roll Damping By fitting these predicted bilge keel components a simple prediction formula is deduced as follows. x1  B d , x 2  C b , x3  C m , x 4  OG d , x5  ωˆ x 6   a (deg) , x 7  bBK B , x8  l BK LPP B Bˆ BK  ABK  exp( B BK 1  B BK 2  x3 BK 3 )  x5 ABK  f 1 ( x1 , x 2 )  f 2 ( x 6 )  f 3 ( x 7 , x8 ) f 1  ( 0.3651x 2  0.3907)  ( x1  2.83) 2  2.21x 2  2.632 2

f 2  0.00255 x 6  0.122 x 6  0.4794 2

f 3  ( 0.8913 x 7  0.0733 x 7 )  x8

2

2

 (5.2857 x 7  0.01185 x 7  0.00189)  x8 B BK 1  {5 x 7  0.3 x1  0.2 x8 2

 0.00125 x 6  0.0425 x 6  1.86}  x 4 B BK 2  15 x 7  1.2 x 2  0.1x1 2

 0.0657 x 4  0.0586 x 4  1.6164 B BK 3  2.5 x 4  15.75  0.5  C b  0.85, 2.5  B / d  4.5      1.5  OG / d  0.2, 0.9  C m  0.99    0 . 01  /  0 . 06 , 0 . 05  /  0 . 4 b B l L BK BK pp  

(38)

5 Validation of Proposed Method To verify the validity of the proposed method, the values of the roll damping calculated by the original Ikeda’s method and the proposed method are compared on two kinds of ships. The results of these comparisons are shown in Figure 14.

A Simple Prediction Formula of Roll Damping of Conventional Cargo Ships

483

These results indicate that the calculated values of the roll damping by the proposed formulas are in good agreement with the roll damping calculated by the Ikeda’s method, although the estimated values by using the proposed method have some errors about 10% when ˆ becomes larger for the ship with OG/d=-0.2 as shown in Figure 14. However, the error margin becomes small in the lowfrequency range because the bilge keel component is predominant.

(a) OG/d = -0.2

(b) OG/d =-1.5 Fig. 14 Comparison between Ikeda’s method and proposed one of roll damping at L/B=6.0, B/d=4.0, Cb=0.65, Cm=0.98, φa=10°, bBK/B=0.025 and lBK/Lpp=0.2.

The results in Figure 14 suggest that the errors come from the discrepancy of the wave component between the proposed method and the potential theory. Especially, this discrepancy grows larger on condition of B/d=4.5 and OG/d=-0.2 ~ -0.5, so it should be noted for the use of the proposed method on this condition. This may be because the wave component of the roll damping intricately depends on frequency and locations of roll axis. Therefore, if more accurate prediction is needed, the calculated wave damping by any potential theory should be used.

484 Y. Kawahara et al.

6 Limitation of Ikeda’s Method In recent years, the number of ships that have buttock flow stern, such as large passenger ship or pure car carrier, has been increasing. In such a type of ships, however, prediction accuracy of the roll damping calculated by Ikeda’s method might decrease remarkably. In order to investigate its limitation, free roll decay tests are carried out by using three types of model ships, large passenger ship (LPS), modern pure car carrier (PCC) and wide breadth and shallow draft car carrier (WSPCC). The principal particulars of these ships and the body plan of a large passenger ship are shown in Table 1 and Figure 15, respectively. As shown in Figure 15, the ship has shallow stern-bottom with something like a skeg. Table 1. Principal particulars of the ships Scale LOA(m) LPP(m) Breadth(m) Draft(m) Displacement(kg) GM(m) Tnr(sec) OG/d BBK/Breadth LBK/LPP

LPS 1/125 2.200 1.933 0.287 0.067 26.98 0.0126 1.88 -1.13 -

PCC 1/96 2.083 2.00 0.336 0.0938 33.43 0.0194 2.06 -0.071 0.0217 0.225

WSPCC 1/96 2.083 2.00 0.378 0.0772 28.05 0.063 1.05 -1.35 0.0217 0.225

Fig. 15 Body Plan of large passenger ship.

In Figure 16, the obtained results of the extinction coefficient N are shown. The results demonstrate that the accuracy of Ikeda’s prediction method decrease remarkably when the roll angle is small. However, the error of roll damping becomes smaller for the large passenger ship and the pure car carrier in condition of large roll angle. For the wide breadth and shallow draft car carrier, however, the roll damping calculated by Ikeda’s prediction method is overestimated from the experimental result for whole roll angle. This is pointed out by Tanaka et al. (1981) that the

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485

effect of the bilge keel component on the roll damping significantly decreases for the wide breadth and shallow draft ships. Also, the discrepancies between the simple prediction formula and Ikeda’s method are attributed to differences between methodical series model and real model ship and to using out of its possible calculating condition. Then, the extinction coefficient N of a large passenger ship is relatively large even for naked hull because of relatively small bilge radius. Thus, Ikeda’s prediction method is valid only in large roll angle for a modern type of ships with buttock flow stern, and overestimate the roll damping for a very flat ship with large bilge keels.

(a) Large passenger ship

(b) Pure car carrier

(c) Wide breadth and shallow draft car carrier Fig. 16 Comparison of roll damping between Ikeda’s method and experimental result

7 Conclusions A simple prediction method of the roll damping of ships is developed on the basis of the Ikeda’s original prediction method which was developed in the same concept as a strip method for calculating ship motions in waves. Using the data of a ship, B/d,

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Cb, Cm, OG/d, ˆ , bBK/B, lBK/LPP, φa, the roll damping of a ship can be approximately predicted. Moreover, the limit of application of Ikeda's prediction method to modern ships that have buttock flow stern is demonstrated by the model experiment. The computer program of the developed method can be downloaded from the Home Page of Ikeda’s Labo. (http://www.marine.osakafu-u.ac.jp/~lab15/roll_damping.html). The computer program of Ikeda’s roll damping prediction method can be downloaded from the Home page of Katayama’s Labo. (http://www.marine.osakafu-u.ac.jp/~lab07/katalaboHP/New_program_of_roll_damping/Roll%20Damping%20Program.html).

Acknowledgements This work was supported by a Grant-in Aid for Scientific Research of the Japan Society for Promotion of Science (No. 18360415). The authors wish to express sincere appreciation to Prof. N. Umeda of Osaka University for valuable suggestions to this study.

References Himeno Y (1981) Prediction of ship roll damping, state of the art. Rep of Dept of Nav Archit & Mar Eng, Univ of Michigan 239 Ikeda Y, Himeno Y, Tanaka N (1976) On roll damping force of ship, effects of friction of hull and normal force of bilge keels. J of the Kansai Soc of Nav Archit, Japan 161 41:49 Ikeda Y, Komatsu K, Himeno Y, Tanaka N (1977a) On roll damping force of ship, effects of hull surface pressure created by bilge keels, J of the Kansai Soc of Nav Archit, Japan 165 31:40 Ikeda Y, Himeno Y, Tanaka N (1977b) On eddy making component of roll damping force on naked hull, J of the Soc of Nav Archit, Japan 142 59:69 Ikeda Y, Himeno Y, Tanaka N (1978) Components of roll damping of ship at forward speed, J of the Soc of Nav Archit, Japan 143 121:133 Ikeda Y, Fujiwara T, Himeno Y, Tanaka N Velocity field around ship hull in roll motion, J of the Kansai Soc of Nav Archit Japan 171 33:45 Ikeda Y (1982) Prediction method of roll damping, Rep of Dept of Nav Archit, Univ of Osaka Prefect Ikeda Y. (1984) Roll damping, 1ST Symp of Mar Dyn Res Group Japan 241:250 Kawahara Y, (2008) Characteristics of roll damping of various ship types and a simple prediction formula of roll damping on the basis of Ikeda’s method, The 4th Asia-Pacific Workshop on Mar Hydrodymics, Taipei 79:86

A Study on the Characteristics of Roll Damping of Multi-Hull Vessels Toru Katayama, Masanori Kotaki, Yoshiho Ikeda Osaka Prefecture University

Abstract In this study, for a catamaran and a trimaran as multi-hull vessels, the characteristics of the roll damping are investigated experimentally. A free roll decay test and a forced roll motion test with and without forward speed are carried out. The results show that the roll damping of them is much larger than that of conventional monohull vessels, and the component created by side-hull accounts for a significant rate for trimaran. Especially, at the condition without forward speed, the interference of waves created by hulls are significant, the measured roll damping values by different experiments are different on the basis of different water surface condition created by hulls. Moreover, the simplified prediction method is proposed.

1 Introduction They are very important to understand the characteristics of the roll damping for any kind of vessels and to estimate it adequately, because it is significantly affect on the occurrence of parametric rolling, the amplitude of resonances and so on. However, it is very complicated to calculate the roll damping theoretically, because of significant viscous component depending on vortex shedding. It is known that there is a prediction method of the roll damping proposed by Ikeda (one of authors) for conventional displacement type of mono-hull vessels, barge ships and a small hard chine fishing boat. It is composed of wave making, friction, transverse lift and eddy making prediction component those are developed with theoretically and experimentally backgrounds based on the hydrodynamic characteristics of the roll damping for the above-mentioned types of vessels. Therefore, it is difficult to apply the method to the resent vessels that have large different hull form from above-mentioned types of vessels. In this study, for catamaran and trimaran as multi-hull vessels, the characteristics of the roll damping are investigated experimentally. And based on the results, the simplified prediction method is proposed. M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_27, © Springer Science+Business Media B.V. 2011

487

488 T. Katayama et al.

2 Models The photographs of two model ships are shown in Figs. 1 and 2, and their principle particulars are shown in Tables 1 and 2. The model of catamaran in Fig. 1 is wave piercing type high-speed craft, and its loading conditions are decided by

Fig. 1 Model of catamaran.

Fig. 2 Model of trimaran and its hull arrangement.

A Study on the Characteristics of Roll Damping of Multi-Hull Vessels 489

based on over-100m class wave piercing catamaran. On the other hand, the model of trimaran in Fig. 2 is stabilized slender mono-hull type high-speed craft, and its distributions of displacement for main-hull and side-hulls are decided by based on a real high speed vehicle-passenger trimaran ferry. Table 1 Principal particulars of trimaran. -

Main-hull

L/B

12

Side-hull 10.2

LOA (m)

1.5

0.888

Lpp (m)

1.42

0.808

breadth (m)

0.125

0.0874

depth (m)

0.1

0.07

draft (m)

0.037

0.012

displacement (N)

30.8

0.778×2

GM (m)

0.163

0.163

Tn (sec)

1.04

0.104

Table 2 Principal particulars of catamaran. Scale

1/80

LOA (m)

1.408

LPP (m)

1.32

breadth( m)

0.38

depth (m)

0.07

draft (m)

0.04

displacement (N)

20.9×2

GM (m)

0.85

Tn (sec)

0.46

3 Characteristics of Roll Damping without Forward Speed In order to investigate the characteristics of roll damping of the models, at first, a free roll decay test is carried out without forward speed. In this test, for model, heaving, pitching, rolling, swaying and yawing are free, and it is initially heeled by fixing only rolling at arbitrary angle. From the condition with which model’s attitude is balancing, motions are measured after freeing rolling. The measured damping curves of rolling for each model are shown in Figs. 3 and 4. The result for trimaran in Fig. 3 seems to continuously damp, on the other hand, the result for catamaran in Fig. 4 shows that the damping curve has some different periods and it damps irregularly.

490 T. Katayama et al.

Therefore, in this study, the results of the free roll decay test with catamaran, only the first swing is analyzed. And the results of the same test with trimaran are analysed for the first swing and other swings separately. ROLL 4 deg. t(sec)

0 0 –4

Fig. 3 Time histories of roll motion of the trimaran.

ROLL 4 deg. 0 0

4

t(sec) 8

–4 –8

Fig. 4 Time histories of roll motion of the catamaran.

As another measurement of the roll damping, a forced roll motion test is also carried out. In the analysis, one degree of freedom of rolling motion equation is used, in which nonlinear terms are replaced by equivalent linear terms

( I 44  a 44 )  B44  C 44  M R ,

(1)

where (I44 + a44), B44, C44 and MR denote the inartia term, damping term, restoring term and roll excitation moment, respectively. In this test, model is forced to sinusoidal rolling around the center of gravity as

   a sin t ,

(2)

A Study on the Characteristics of Roll Damping of Multi-Hull Vessels 491

The measured roll moment is represented with Fourier series. Using the Fourier coefficient of the fundamental period for measured MR, the amplitude MRF in phase of roll angular velocity of measured MR is obtained, and B44 is written as follows

B44 

M RF

a

,

(3)

where ω and a are roll circular frequency and amplitude. The roll damping coefficient B44 can be represented by Bertin’s N-coefficient on the condition that the energy losses of them in one period are equal,

B44  2 g

GM  a



N,

(4)

where  is displacemental volume, and the unit of a is degree. And B44 is nondimensionalized as follow B44 Bˆ 44  B 2

B , 2g

(5)

where B is breadth of vessel.

a=2(deg)

a=5(deg)

a=3(deg)

a=5(deg)

exposed

a=6(deg)

a=10(deg)

a=10(deg)

a=16(deg)

Fig. 5 Relative position of the heeling model to the water surface.

492 T. Katayama et al.

In Fig. 5, the relative positions of the models to the water surface when the model has heel angle. In the free roll decay test, the initial heel angles, where hull of model is not exposed the air, are selected. And the experimental conditions in forced roll motion test are shown in Table 3. Table 3 Experimental conditions in forced roll motion test. -

trimaran

center height of roll motion (m)

0.156

Catamaran 0.144

trim angle (deg)

0

0

forced rolling period: T (sec)

0.4, 0.8, 1.04, 1.3

0.3, 0.46, 0.7, 1.0

roll amplitude: a (deg)

2, 5, 10

3, 5, 10

In Figs. 6 ~ 9, B44 and N-coefficient for two models are shown. From the results of N-coefficient for both models in Figs. 6 and 7, it is confirmed that their roll damping are sensuously large in comparison with it for common mono-hull vessels and the N10 is about 0.08. On the other hand, from the results of B44, the effect of amplitude is observed. And, from Figs. 8 and 9, the effect of roll period is also clearly found. On the comparisons among different tests and analysis, quantitatively slightly different results are obtained. The results for trimaran show that the results of the first swing of the free roll decay test indicate lager value than those of the other swing of the same test and the forced roll motion test. For the results for catamaran, the same N coefficient free roll decay test(first swing) free roll decay test(2nd～5th swings)

0.4

forced roll motion test Wave making component +Eddy making component Wave making component Main hull roll damping

0.2

0 0

5

10 15 Roll amplitude(deg)

5

10 15 Roll amplitude(deg)

0.06 B44 hat 0.04 0.02 0 0

Fig. 6 Roll damping of the trimaran.

A Study on the Characteristics of Roll Damping of Multi-Hull Vessels 493

tendency is also observed. As a reason of this, it is supposed that the free surface conditions in these tests are different and there is the disturbed water surface by previous swing excepting the results of the first swing of free roll decay test. N coefficient

free roll decay test(first swing) free roll decay test(2nd～5th swings)

0.4

forced roll motion test Wave making component +Eddy making component Wave making component

0.2

0 0

5

10 15 Roll amplitude(deg)

5

10 15 Roll amplitude(deg)

0.06 B44hat 0.04 0.02 0 0

Fig. 7 Roll damping of the catamaran.

Forced roll motion test amplitude(deg) :2 :5 : 10

0.06 B44hat

: Wave making component Wave making component +Eddy making component amplitude(deg) :2 :5

0.04 0.02 0 0

0.4

0.8

1.2 roll period(sec)

Fig. 8 Roll damping of the trimaran.

: 10

494 T. Katayama et al. Forced roll motion test amplitude(deg) :3 :5 : 10 : Wave making component Wave making component +Eddy making component amplitude(deg) :3 :5

: 10

0.06 B44hat 0.04 0.02 0 0

0.4

0.8

1.2 roll period(sec)

Fig. 9 Roll damping of the catamaran.

As reasons for the effect of amplitude on B44, the nonlinearity of wave component caused by large amplitude motion, the interference of wave among hulls and the viscosity effects are occurred to. In Figs. 8 and 9, the effect is also clearly observed for small amplitude. Therefore the viscosity component is considered in this study. Fig. 10 shows significant roll damping components acting on catamaran. One of them is the wave making component BW which is created by the almost vertically oscillating demi-hull. This component can be roughly estimated by using the heave potential damping of demi-hull B33 as following equation BW   BW  a  2l B 33 l a  2l 2 B 33 ,

(6)

where l is the distance of demi-hull from the centre line shown in Fig. 10. It is noted that the B43 is not include the interference of wave among hulls, in this study. Another component is the eddy making component and this component acts on the demi-hull moving upward shown in Fig. 10. This component can be expressed by using drag coefficient CDA

1 SC DA l a 2 2 8 a  l2 SC DA  a  , 3 8 a  l2 SC DA 3

B E  

(7)

where  is density of fluid, S is hull projected area from below. In this study, CDA is obtained from the data in Fig. 11 and following equation

A Study on the Characteristics of Roll Damping of Multi-Hull Vessels 495

C DA  C D  C DF ,

(8)

In Fig. 11, CD is the total drag coefficient of a 2-D section for a c/t, and CDF and CDA are the drag coefficients acting on forward-body and aft-body of the 2-D section, respectively. And CDF is the drag coefficient at c/t=∞. Moreover, CDA is including roughly KC number effects by using the ratio of CD for a KC number shown in Fig. 12. In this study, KC number is assumed the following equation KC 

2l a  2l a d   Rn  ,  b  

(9)

where b is the breadth of cross section of demi-hull. For catamaran, the total roll damping is calculated by adding these two components. On the other hand, for trimaran, the total roll damping is calculated by adding these two components and the main-hull’s roll damping calculated by Ikeda’s prediction method1). In Figs. 6 ~ 9, the estimated B44 is also shown as lines. The estimation can express the effects of roll amplitude and period for the trimaran, however, can not always for the catamaran. As one of reasons for the difference, it is supposed that the interference of waves among hulls is significant for the catamaran.

Fig. 10 Roll damping components for catamaran

Fig. 11 Drag coefficients of 2-D section with several aspect ratio c to t. In this study, c is draft and t is breadth of demi-hull. (S.F. Hoerner, 1965)

496 T. Katayama et al.

Fig. 12 KC number‘s effects of drag coefficient for cylinder. (Kuelegan G.H., Carpenter L.H., 1958)

4 Characteristics of Roll Damping with Forward Speed In order to investigate the characteristic of the roll damping with forward speed, the forced roll motion test is carried out. In this test, the cases of forward speed are decided by based on the maximum speed of some real multi-hull high-speed craft. The experimental condition is shown in Table 4. Table 4 Experimental conditions in forced roll motion test

Fn trim angle (deg) forced rolling period: T (sec) roll amplitude: a (deg)

trimaran

catamaran

0.148, 0.296

0.136, 0.272

0.443, 0.591

0.407, 0.543

0

0

0.4, 0.8

0.3, 0.46

1.04, 1.3

0.7, 1.0

2, 5, 10

3, 5, 10

In Figs. 13 and 14, B44 for two models are shown, and their forced roll periods are their roll natural periods, respectively. For the results of the trimaran in Fig. 13, the effect of forward speed on B44 is small. On the other hand, for the catamaran in Fig. 14, the effect is significant. In Figs. 15 and 16, the effects of roll period on B44 for two models are shown, and their Fn are their service speed, respectively. The result of the trimaran in Fig. 15 shows the same tendency as the results at Fn=0. On the other hand, the result of the catamaran in Fig. 16 shows the different tendency from the results at Fn=0.

A Study on the Characteristics of Roll Damping of Multi-Hull Vessels 497

0.06 B44hat

roll amplitude(deg) :2 :5 : 10

0.04 0.02 0 0

Fn 0.2

0.4

0.6

Fig. 13 Roll damping of the trimaran at the roll natural period

0.06 B44hat

roll amplitude(deg) :3 :5 : 10

0.04 0.02 0 0

Fn 0.2

0.4

0.6

Fig. 14 Roll damping of the catamaran at the roll natural period

0.06 B44hat

roll amplitude(deg) :2 :5 : 10

0.04 0.02 0 0

0.4

0.8

1.2 roll period(sec)

Fig. 15 Roll damping of the trimaran at Fn=0.591

498 T. Katayama et al.

0.06 B44hat

roll amplitude(deg) :3 :5 : 10

0.04 0.02 0 0

0.4

0.8

1.2 roll period(sec)

Fig. 16 Roll damping of the catamaran at Fn=0.543

5 Conclusion In this study, in order to investigate the characteristics of the roll damping for multihull vessel, the roll damping of a catamaran and a trimaran is measured. Some conclusions can be remarked as follow. 1. In the result of free roll decay tests, time histories of roll motion of the catamaran damp irregularly. This is caused by the interference of waves created by two hulls. 2. The roll damping of a multi-hull has effects of roll amplitude and period. The effects for a trimaran can be created by the eddy making component created by the demi-hull moving upward. However, for a catamaran, it may be necessary to take into account the interference of waves created by hulls. 3. The effect of forward speed on the roll damping for a trimaran is small. On the other hand, effect for a catamaran is significant. 4. The characteristics of the roll damping of a catamaran is different from that of a trimaran at Fn=0.

References Ikeda Y (1984) Roll Damping. The 1th Symp of Mar Dyn Res Group 241-249 Motora S, Koyama T, Fujino M et al (1997) Dyn of Ships and Offshore Struct. S F Hoerner (1965) FLUID-DYNAMIC DRAG. Publ by the Author Kuelegan G H, Carpenter L H (1958) Forces on cylinders and plates in an oscillating fluid. J. Res Nat Bur Stand LX N 5:423-440

7 Probabilistic Assessment of Ship Capsize

Capsize Probability Analysis for a Small Container Vessel E.F.G. van Daalen*, J.J. Blok* H. Boonstra** *MARIN, Wageningen **Delft University of Technology, The Netherlands

Abstract In this paper we investigate the long term capsize probability for a small container vessel which is operated in a regular service schedule on the North Sea and the north-east part of the Atlantic Ocean. The numerical simulation techniques involved are a ship route scenario simulation method and a time domain simulation method for large amplitude ship motions. From the combined statistics of the scenario simulations and the ship motion simulations we derive the capsize probability. Finally, the effect of avoiding bad weather conditions on the short term and long term capsize probability is discussed.

1 Introduction In this paper we investigate the long term capsize probability for a small container vessel which is operated in a regular service schedule on the North Sea and the north-east part of the Atlantic Ocean. The numerical simulation techniques involved are a ship route scenario simulation method and a time domain simulation method for large amplitude ship motions. The scenario simulation tool GULLIVER was developed at MARIN as a product for clients interested in the performance of their ship(s) in service conditions, with respect to safety, economy and reliability. The large amplitude ship motion program FREDYN was developed at MARIN within the framework of the Cooperative Research Navies project. Both methods have been applied successfully in many projects and have proven their value in both a research and a commercial context. Our primary goal is to find a way to combine these two tools into a method for calculating the long term capsize probability. With GULLIVER we are able to account for the effect of involuntary speed loss due to waves, wind and current and to quantify the encountered weather conditions in terms of (long term) scatter diagrams. This part of the approach is described in Section 2. With FREDYN we are able to identify the conditions in which the ship is prone to capsize. This part of the work is described in Section 3. Combining the two gives an impression of the average short term capsize risk. The final step is to translate this short term capsize risk into a long term capsize risk, which is described in Section 4. As for the captain’s influence on the safety of the ship, Gulliver provides us some means to quantify the effect of the captain’s decisions based on the ship behaviour. M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_28, © Springer Science+Business Media B.V. 2011

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However, at the time of writing this paper, these measures are restricted to reducing speed whenever a certain criterium is exceeded; they do not (yet) include the possibility to change course during a trip. In Section 5 we will present a method to mimic course-deviations a posteriori, in order to estimate the effect of the captain’s decisions on the capsize risk.

start: set initial position

environment databases

get local environment

calculate sustained speed

RAO database

calculate behaviour

scenario

criteria satisfied ?

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