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  • 1. Contemporary Ideas on Ship Stability and Capsizing in Waves
  • 2. FLUID MECHANICS AND ITS APPLICATIONS Series Editor: R. MOREAU MADYLAM Ecole Nationale Supérieure d’Hydraulique de Grenoble Boîte Postale 95 38402 Saint Martin d’Hères Cedex, France Aims and Scope of the Series fundamental role. As well as the more traditional applications of aeronautics, hydraulics, heat and mass transfer etc., books will be published dealing with topics which are currently in a state of rapid development, such as turbulence, suspensions and multiphase fluids, super and hypersonic flows and numerical modeling techniques. It is a widely held view that it is the interdisciplinary subjects that will receive intense scientific attention, bringing them to the forefront of technological advancement. Flu- ids have the ability to transport matter and its properties as well as to transmit force, therefore fluid mechanics is a subject that is particularly open to cross fertilization with other sciences and disciplines of engineering. The subject of fluid mechanics will be highly relevant in domains such as chemical, metallurgical, biological and ecological engineering. This series is particularly open to such new multidisciplinary domains. The median level of presentation is the first year graduate student. Some texts are mono- graphs defining the current state of a field; others are accessible to final year undergrad- uates; but essentially the emphasis is on readability and clarity. The purpose of this series is to focus on subjects in which fluid mechanics plays a Volume 96 For further volumes:
  • 3. Editors Jean Otto de Kat • Kostas Spyrou • Naoya Umeda Marcelo Almeida Santos Neves • Vadim L. Belenky Contemporary Ideas on Ship Stability and Capsizing in Waves
  • 4. No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any Printed on acid-free paper Springer is part of Springer Science+Business Media ( 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 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2011929340 © Springer Science+Business Media B.V. 2011 of being entered and executed on a computer system, for exclusive use by the purchaser of the work. ISSN 0926-5112 e-ISBN 978-94-007-1482-3ISBN 978-94-007-1481-6 Marcelo Almeida Santos Neves Department of Naval Federal University of Rio de Janeiro Parque Tecnologico, 21941-972 Rio de Janeiro Brazil Jean Otto de Kat Esplanaden 50 1098 Copenhagen Denmark Vadim L. Belenky Naval Surface Warfare Center Carderock D 20817-5700 West Bethesda Maryland USA Kostas Spyrou School of Naval Architecture National Technical University of Athens Iroon Polytechnion, 157 73 Athens Greece and Marine Naoya Umeda Department of Naval Osaka University Graduate School of Eng Yamadaoka Suita 2-1 565-0971 Osaka Japan DOI 10.1007/978-94-007-1482-3 MacArthur Blvd 9500 Editors A. P. Moeller - Maersk A/S Cover design: eStudio Calamar S.L. Architecture and Oce Quadra 7 Architecture and Oce Zographou 9 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.
  • 5. v 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 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 A. Yucel Odabasi and Erdem Uçer
  • 6. Table of Contentsvi 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 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 A. Rodríguez
  • 7. 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
  • 8. Table of Contentsviii 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
  • 9. Table of Contents ix 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
  • 10. 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) xi
  • 11. List of Corresponding Authors 1 Stability Criteria Review of Available Methods for Application to Second Level Vulnerability Criteria Vadim Belenky A Basis for Developing a Rational Alternative to the Weather Criterion: Problems and Capabilities K.J. Spyrou Conceptualising Risk Dracos Vassalos Evaluation of the Weather Criterion by Experiments and its Effect to the Design of a RoPax Ferry Shigesuke Ishida Evolution of Analysis and Standardization of Ship Stability: Problems and Perspectives Yury Nechaev SOLAS 2009 – Raising the Alarm Dracos Vassalos 2 Stability of the Intact Ship Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas Historical Roots of the Theory of Hydrostatic Stability of Ships Horst Nowacki xiii E. Uçer
  • 12. List of Corresponding Authorsxiv The Effect of Coupled Heave/Heave Velocity or Sway/Sway Velocity Initial Conditions on Capsize Modeling Leigh S. McCue The Use of Energy Build Up to Identify the Most Critical Heeling Axis Direction for Stability Calculations for Floating Offshore Structures Joost van Santen Some Remarks on Theoretical Modelling of Intact Stability Naoya Umeda 3 Parametric Rolling An Investigation of Head-Sea Parametric Rolling for Two Fishing Vessels Marcelo de Almeida Santos Neves Simple Analytical Criteria for Parametric Rolling K.J. Spyrou Experimental Study on Parametric Roll of a Post-Panamax Containership in Short-Crested Irregular Waves Hirotada Hashimoto Model Experiment on Parametric Rolling of a Post-Panamax Containership in Head Waves Harukuni Taguchi Numerical Procedures and Practical Experience of Assessment of Parametric Roll of Container Carriers Vadim Belenky Parametric Roll and Ship Design Riaan van’t Veer
  • 13. List of Corresponding Authors xv Parametric Roll Resonance of a Large Passenger Ship in Dead Ship Condition in All Heading Angles Toru Katayama Parametric Rolling of Ships – Then and Now J. Randolph Paulling 4 Broaching-to Parallels of Yaw and Roll Dynamics of Ships in Astern Seas and the Effect of Nonlinear Surging K.J. Spyrou Model Experiment on Heel-Induced Hydrodynamic Forces in Waves for Realising Quantitative Prediction of Broaching Hirotada Hashimoto Perceptions of Broaching-To: Discovering the Past K.J. Spyrou 5 Nonlinear Dynamics and Ship Capsizing Use of Lyapunov Exponents to Predict Chaotic Vessel Motions Leigh S. McCue Applications of Finite-Time Lyapunov Exponents to the Study of Capsize in Beam Seas Leigh S. McCue Nonlinear Dynamics on Parametric Rolling of Ships in Head Seas Marcelo A.S. Neves, 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
  • 14. List of Corresponding Authorsxvi A Study on the Characteristics of Roll Damping of Multi-hull Vessels Toru Katayama 7 Probabilistic Assessment of Ship Capsize Capsize Probability Analysis for a Small Container Vessel E.F.G. van Daalen Efficient Probabilistic Assessment of Intact Stability K.J. Spyrou Probability of Capsizing in Beam Seas with Piecewise Linear Stochastic GZ Curve Vadim Belenky Probabilistic Analysis of Roll Parametric Resonance in Head Seas Vadim L. Belenky 8 Environmental Modeling Sea Spectra Revisited Maciej Pawłowski On Self-Repeating Effect in Reconstruction of Irregular Waves Vadim Belenky New Approach To Wave Weather Scenarios Modeling Alexander B. Degtyarev 9 Damaged Ship Stability Effect of Decks on Survivability of Ro–Ro Vessels Maciej Pawłowski
  • 15. 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 Exploring the Influence of Different Arrangements of Semi-Watertight Spaces on Survivability of a Damaged Large Passenger Ship Riaan van’t Veer Time-Based Survival Criteria for Passenger Ro-Ro Vessels Dracos Vassalos Pressure-Correction Method and Its Applications for Time-Domain Flooding Simulation Pekka Ruponen 10 CFD Applications to Ship Stability Applications of 3D Parallel SPH for Sloshing and Flooding Dracos Vassalos Simulation of Wave Effect on Ship Hydrodynamics by RANSE Dracos Vassalos A Combined Experimental and SPH Approach to Sloshing and Ship Roll Motions Luis Pérez-Rojas 11 Design for Safety Design for Safety with Minimum Life-Cycle Cost Dracos Vassalos
  • 16. List of Corresponding Authorsxviii 12 Stability of Naval Vessels Stability Criteria Evaluation and Performance Based Criteria Development for Damaged Naval Vessels Andrew J. Peters A Naval Perspective on Ship Stability Arthur M. Reed 13 Accident Investigations New Insights on the Sinking of MV Estonia Dracos Vassalos Experimental Investigation on Capsizing and Sinking of a Cruising Yacht in Wind Naoya Umeda
  • 17. 1 Stability Criteria
  • 18. M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, 3 Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_1, © Springer Science+Business Media B.V. 2011 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 state- of-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
  • 19. 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). C.C. Bassler et al.4
  • 20. Review of Available Methods for Application to Second Level Vulnerability Criteria 5 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. 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). Stability Failure Mode Vulnerability Criteria Level 1 Ship Design Pass Fail Vulnerability Criteria Level 2 Pass Fail IMO IS Code Performance-Based CriteriaFor each mode Fig. 1. The proposed assessment process for next generation intact stability criteria IMO IS Code
  • 21. 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). 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 balance- based 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. 2. Capsizing due to pure-loss of stability Fig. 3. 0 50 100 150 50 50 100 150 200 t, s Time history of roll and capsizing Phase of encounter wavePhase of heeling moment 0 10 20 30 40 50 60 0.5 1 Wave trough Calm water Wave crest Magnitude of heeling moment -0.5  deg Righting / Heeling Arm, m C.C. Bassler et al.6 A concept of an effective wave
  • 22. Review of Available Methods for Application to Second Level Vulnerability Criteria 7 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 up- crossing 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
  • 23. 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). 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). Fig. 4. Partial (a) and total (b) stability failure caused by parametric resonance 0 100 200 300 400 500 t, s -20 20 , deg 0 100 200 300 400 500 t, s -50 50 100 150 200 , degb) a) C.C. Bassler et al.8
  • 24. Review of Available Methods for Application to Second Level Vulnerability Criteria 9 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).
  • 25. 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). 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 Fig. 5. Sample records of simulated parametric roll in long-crested seas (Shin, et al., 2004) Fig. 6. Distribution of roll angles during parametric resonance in (a) long-crested and (b) short crested seas 0 200 400 600 800 1000 1200 1400 -40 -20 0 20 40 Time, s RollAngle,deg Time, s 0 200 400 600 800 1000 1200 1400 -40 -20 0 20 40 RollAngle,deg a) b) C.C. Bassler et al.10
  • 26. Review of Available Methods for Application to Second Level Vulnerability Criteria 11 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 Monte- Carlo 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 &
  • 27. 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. 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 surf- riding can be realised only in the vicinity of the trough and, in fact, only if sufficient rudder control has been applied. Fig. 7. Changing of surging/surf-riding behavior with increasing nominal Froude number G x Fn Periodic surging Surf-riding equilibria At wave trough At wave crest Fncr1 Fncr2 cos(2xG/ a b c d C.C. Bassler et al.12
  • 28. Review of Available Methods for Application to Second Level Vulnerability Criteria 13 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 state- space 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. 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). Fig. 8. Phase plane with surging and surf-riding, Fncr1<Fn<Fncr2 300 200 100 0 100 10 10 20 G, m G, m/s . Surf-riding Surf-riding Surging Surging
  • 29. 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 surf- riding; 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 unlikely to broach the nature of such a broaching mechanism g. It has been shown that, a 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. Fig. 9. Phase plane with surf-riding only Fn>Fncr2 are through surf-riding. Let’s consider now , which we will call “direct broaching” since the ship is not required to go through surf-ridin 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). 300 200 100 0 100 10 10 20 G, m Surf-riding Surf-riding Surf-riding G, m/s . C.C. Bassler et al.14
  • 30. Review of Available Methods for Application to Second Level Vulnerability Criteria 15 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 Fig. 10. Direct broaching One period motion Stable sub- harmonic yaw One period motion Stable sub- harmonic yawFlip bifurcation Fold bifurcation One period motion Amplitudeofperiodicmotion C commanded heading Rudderangle Yaw angle
  • 31. 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. 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 Fig. 11. ith increasing of wave amplitude 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 SMOOTH BOUNDARY FRACTAL BOUNDARY AreaofSAFEBASIN 1 FINAL CRISIS Normalized amplitude of excitation a b c d e f C.C. Bassler et al.16 Deterioration of safe basin w
  • 32. Review of Available Methods for Application to Second Level Vulnerability Criteria 17 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. 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 Fig. 12. Separation principle in a piecewise linear method  t Capsized equilibrium Threshold Upcrossing leading to capsizing Upcrossing not leading to capsizing
  • 33. 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 C.C. Bassler et al.18
  • 34. Review of Available Methods for Application to Second Level Vulnerability Criteria 19 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 long- term approach using a scatter diagram for a representative part of the World Ocean. 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. 9 Acknowledgments
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  • 37. 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 C.C. Bassler et al.22
  • 38. Review of Available Methods for Application to Second Level Vulnerability Criteria 23 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. 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
  • 39. M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, 25 Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_2, © Springer Science+Business Media B.V. 2011 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 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 Abstract
  • 40. 26 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 2247.15.1  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 work done by the wind excitation as the ship rolls from the wind-side to the lee- side should not exceed the potential energy at the limiting angle θ2. degrees which can be assumed as an explicit safety limit whichever of the three is the lowest. Expressed as an energy balance: the K.J. Spyrou
  • 41. A Basis for Developing a Rational Alternative to the Weather Criterion 27 Fig. 1 The IMO weather criterion 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 θ1 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 θ1 both in terms of hull dimensions, form θ θ1 θ0 θ2 θc lw1 lw2 θ0 Restoring and heeling lever b a θ θ2 , it is not considered during the final half-cycle
  • 42. 28 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). 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 . 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 12 4.1 AA  . θ θdownflooding 25 deg GZC c GZmax 0 Fig K.J. Spyrou
  • 43. 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 time- dependent 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:
  • 44. 30  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      τxεxfτx ,g (1) Where: x is in our case the vector   ddxx /, of roll displacement (normalised with respect to the vanishing angle) and roll velocity Fig. 4 The safe basin and the distance of manifolds  xf represents the “unperturbed” Hamiltonian part of the dynamical system; and  ,xg is the “perturbation” which includes the damping and the explicitly time-dependent wave-forcing.  is a parameter that represents the smallness of  ,xg . It can be shown that the closeness of manifolds is expressed as (Fig. 4):        20 0 ε xf τε τ O Μ d  (2) d t( ) x f x( ) Hamiltonian K.J. Spyrou
  • 45. A Basis for Developing a Rational Alternative to the Weather Criterion 31 Where 0τ is a phase angle in the range  /2τ0 0  . Since the denominator is of the order of 1 the function  0τM (which is called Melnikov function) is to first order a good measure of the distance of manifolds:         τττ,τxgτxfτ 00 dM     (3) The wedge symbol  means to take the cross product of the vectors f and g . 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:        32 11(1 axxaxaxxxxR  (4) The parameter a indicates the strength of the bias assumed here to be due to beam wind loading. Letting 1xx  and 2xx  the roll motion could be described by the following pair:     20 3 1 2 112 21 βττΩsin1 xFaxxaxx xx   (5) If the bias is relatively strong the unperturbed part is represented by the vector:             3 1 2 11 2 1 xf axxax x (6) Where
  • 46. 32        2 1 x x x (7) While the perturbation is:          2β-Ωτsin 0 xF xg (8) After substitution into (3) the Melnikov function takes the form of the following integral:        DE MMdxFxM     0 - 2020 ττβττsinΩ (9) where  0τEM is the part due to wave forcing that is explicitly time-dependent; and DM is the part due to the damping. 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 2x 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 1 xs the equation of the unperturbed system becomes: F a 0.055 Ω=0.8 Ω=0.9 Ω=1 0.940.920.90.880.860.840.820.8 0.035 0.04 0.045 0.05 K.J. Spyrou
  • 47. A Basis for Developing a Rational Alternative to the Weather Criterion 33    sasaas d sd 112 τ 23 2 2  (10) With some manipulation the above can be written as:   22 qpshs d ds   (11) The parameters that appear in (11) are defined as follows:      a aa qand a a p a h 3 212 3 122 , 2     (12) The solution of (11) is:       τ1cosh 1ττ 22 aqp qp xs    ∓ (13) Differentiation of (13) yields:      2 22 2 τ1cosh τ1sinh1 aqp aqpaq x d dx d ds     (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:                 q pqp q p pqqpqph F arccoshμsin μπsinh 6ππ arccosh32 β 22 22222 (15) The sustainable wave slope is:   2 ν . Ων φF Ak crit  (16)
  • 48. 34 where νφ is the vanishing angle and xx x II I δ ν   . 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 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     53 1 xccxxxR  (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 ( 0c ) as well as softening ( 0c ) 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 maxGZ . The main purpose of the study was to confirm that there is indeed GZ K.J. Spyrou
  • 49. 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.
  • 50. 36 Fig. 7 Integrity diagrams b) Application for an existing ship A ferry was selected which had operated in Greek waters with 153BPL m B = 22.8m T = 6.4m bC = 0.548 and 004.10KG m. The GZ curve was known while the damping as a function of frequency was determined using the well- known 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 integrity 1 0.8 0.6 0.4 0.2 1 0.8 0.6 0.4 0.2 0 0.05 0.1 0.15 0.2 0.01 0.05 0.1 0.2 b 0.25 integrity Forcing amplitude F 0 0.05 0.1 0.15 0.2 0.25 Forcing amplitude F c=1, Ω=1 c=1, Ω=0.88 K.J. Spyrou
  • 51. 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:      zxZxXzxfy ,,  (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  xX which depends only on the nondimensionalised position l x x  where x is the longitudinal position measured from the middle of the ship and l is the entrance length with 11  x . The function  xX is expressed as the 4th order polynomial
  • 52. 38    4 4 3 3 2 21 2 xaxaxa B xX  (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      xn zzxZ  1, (20) where   xtsxn  (21) with s, t free parameters. After a few trials we found that for somewhat realistic hull shapes the parameters 2a 3 and 4a 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 K.J. Spyrou
  • 53. A Basis for Developing a Rational Alternative to the Weather Criterion 39 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 ( 01  z ). 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. Fig. 11 Dynamic stability performance: Left bars are for waves only; while the right bars are for wind and waves. 0,08 0,07 0,06 0,05 0,04 0,03 0,02 0,01 0 1 3 5 7 9 11 S h i p N o 13 15 17 19 0,035 0,03 0,025 0,02 (H/λ)critical 0,015 0,01 0,005 0 1 3 5 7 9 11 S h i p N o 13 15 17 19
  • 54. 40 The free parameters were then varied in a systematic way so that 1432  aaa and ts  . 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. 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 1.5 1.0 0.5 0 0 50 Heel angle (degrees) GZ (m) Area (m-rad) 100 K.J. Spyrou
  • 55. 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 Melnikov- based 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.
  • 56. 42 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. K.J. Spyrou
  • 57. A Basis for Developing a Rational Alternative to the Weather Criterion 43 Spyrou KJ (2000) Designing against parametric instability in following seas. Ocean Eng, 27 653. 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. 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. Appendix I: Melnikov’s Method as Energy Balance Consider again the roll equation with bias:    τcos11β  Faxxxxx (I1) The corresponding Hamiltonian system is:    011  axxxx (I2) with kinetic energy 2 2 1 KE x (I3) and potential energy Fig. I1 Energy balance around the homoclinic orbit 625- 50 Yamagata, M. (1959) Standard of stability adopted in Japan, RINA Transactions, 101, 417-443.
  • 58. 44      432 43 1 2 1 11PE x a x a xcaxxx     (I4) It is obvious that if we take as initial point the saddle (x=1 0x ) 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):   τββDE 2 dxdxx (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      τττsinττsinWE 00 dxFdxF (I6) The time constant 0τ allows to vary the phase between forcing and velocity x . The balance of energies requires therefore that i.e.        τττcosβWEDE 0 2 dxFdx  (I7) which is identical with (9) when set equal to zero i.e. we have obtained a condition identical to the condition   0τ0 M 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 K.J. Spyrou
  • 59. 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 ( )τF 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): ( )2 ε ε OM H M PM H M pH A D s D D s D s + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ −⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ +⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Α = Φ σσ σ (II1) where sH is the significant wave height σ is the RMS value of the Melnikov function ( ) ( ) DE MMM −= 0ττ for unity sH and DM 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 ( )[ ]0τME is DM− . 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 sH 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 sH axis as the critical one. The various quantities that appear in (II1) are calculated as follows: For the damping term DM 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:
  • 60. 46          p q pqqpqphM D arccosh32β 3 1 22222 (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:               00 0 22 2τ dSSdSME FxME  (II3) The spectral density function of the wave forcing is linked to the wave spectrum        SFSF 2 (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        - τ τ π2 1 dexS i x (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 sH ). , , K.J. Spyrou
  • 61. M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, 47 Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_3, © Springer Science+Business Media B.V. 2011 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 “weigh ing” 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. t
  • 62. A. Jasionowski and D. Vassalos48 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.
  • 63. 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) likelihood, chance, probability (b) frequency, rate (c) uncertainty, doubt, randomness (d) risk, hazard, danger, threat (e) analysis, assessment, evaluation (f) risk control measures, risk control options (g) 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.
  • 64. A. Jasionowski and D. Vassalos50 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?
  • 65. 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.
  • 66. A. Jasionowski and D. Vassalos52 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.
  • 67. 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
  • 68. A. Jasionowski and D. Vassalos54 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: (1) Where hzn 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 Principal hazards, hzj Average historical frequency of its occurance, frhz(hzj) 1 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 real- life 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). (2)      jNj n j hzN hzNprhzfrNfr hz  1    NcepwhzNpr kji n k kj i n j iN Hsflood ,, 3 1   
  • 69. 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 derived from (15) for the sea state Hsk, where mHsk 40  , 1 4   Hsk nkHs 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).         30 ln)( 30 ,,,,,, Nt Nc fail Nt kjikjikji fail          (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:              ijcrity ijcritk kji sHs sHsHs   )( 1,, (4)   049.0039.0  critcrity HsHs (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. Fig. 3 The concept of “capsize band”, (Jasionowski et al. 2004), for critical sea states of 0.5 and 4.0m. The concept of a capsize band 6 5 4 3 2 1 0 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 Hs crititcal pdf(Hs) pcap=cdf(Hs) 3 3.5 4 4.5 5
  • 70. A. Jasionowski and D. Vassalos56      1 NtNtNt failfailfail (6)    tNNt failfail 1  (7)    tNNtN evacfail  max (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).
  • 71. Conceptualising Risk 57    ifrNF N Ni NN   max (9) 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).    iFNERisk N i NPLL   max 1 (10)
  • 72. A. Jasionowski and D. Vassalos58 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. (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).    NkNERisk  
  • 73. 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. Fig. 6 Elements of index A, sample case. Fig. 7 Hypothetical evac. completion curve 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 pi si 140120100806040200 0 500 1000 1500 2000 time to evacuaute [minutes] Numberofpassengersevacuated[-] 60 minutes 120 minutes
  • 74. A. Jasionowski and D. Vassalos60 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. 0 500 1000 1500 2000 Number of fatalities evacuation complete 120min evacuation complete 60min 1.00E-01 1.00E-02 1.00E-03 1.00E-04 1.00E-05 1.00E-06 conditionalprobabilitymassfunctionofexactlyN fatalitiesprN (N/hz1 ) Namely in case 1, 1 1 1 1 1 1 2 20.1, 0.3000, 0.9, 0.8560,p s p s    and for case 2, 2 21 2 2 1 1 2 22.1, 0.6000, 0.8, 0.8505,p s p s    were assumed. The resultant F curves are shown in Figure 10 below. N curves
  • 75. 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.
  • 76. A. Jasionowski and D. Vassalos62 Acknowledgements The present research was funded by the EC 6th Framework Programme (2002- 2006), project EC FP6 IP 516278 SAFEDOR SP2.1 . 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 Post- Damage. 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, 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
  • 77. 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):           PLLNEifi ifjfiF N i N N i i j N N i N ij N N i N          max maxmax maxmax 1 1 111 (12) 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: 4 1 max , 1612.0      RangeGZ Ks finali (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 collisionHsF 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).        1612.0 4 max RangeGZ Hscrit (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
  • 78. A. Jasionowski and D. Vassalos64 collisionHs collision e Hs eF    2.116.0 (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).    2.1 lnln16.0 )()( s sHssHs collisioncrit   (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.
  • 79. M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, 65 Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_4, © Springer Science+Business Media B.V. 2011 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
  • 80. 66 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 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. 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; Angle of heel GZb a lw1 lw2 Lever 2 0 1 S. Ishida et al.
  • 81. 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 Lpp [m] 170.0 Breadth B [m] 25.0 Depth D [m] 14.8 Draft D [m] 6.6 Displacement W [ton] 14,983 Blockage coefficient Cb 0.521 B/d 3.79 Area of bilge keels Abk [m2 ] 61.32 Vertical center of gravity KG [m] 10.63 Metacentric height G0M [m] 1.41 Flooding angle f  [deg] 39.5 Natural rolling period Tr [sec] 17.90 Lateral projected area AL [m2 ] 3433.0 Height of center of AL above WL Hc [m] 9.71 Fig. 2 General arrangement
  • 82. 68 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. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 20 40 60 80 Angle of Heel [deg] GZ[m] S. Ishida et al.
  • 83. 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: U B Re    (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: wind wind M M F 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: 21 2 D wind L air L C F C L U A                (3)
  • 84. 70 2 21 2 M L wind air pp C M A U L         (4) 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. /wind wind wind l M F (5) 0 0.25 0.5 0.75 1 1.25 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 Angle of heel (deg.) CD,CL,CM CD CL CM CM (standard criterion) S. Ishida et al.
  • 85. Evaluation of the Weather Criterion 71 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. Fig. 7 Wind heeling moment coefficient for various encounter angles 0.000 0.025 0.050 0.075 0.100 0.125 0.150 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 Angle of heel (deg.) lwind(m) Centre of lateral projected area Angle of heel [deg.] 15 (deg.) 30 45 60 75 90 105 120 135 150 1653530252015105-5-10-15-20-25-30-35 0 0.00 0.25 CM 0.50 0.75
  • 86. 72 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). 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. Half draft CD=0.50.6 0.4 0.2 0.0 −0.2 Angle of Heel (deg.) Iwater/draft −0.4 −0.6 CD=0.6 CD=0.8 CD=1.0 3530252015105−5−10−15−20 0 CD=0.7 CD=0.9 CD=1.1 S. Ishida et al.
  • 87. 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, lw1, was calculated as a function of heel angle (equation (6)). Figs. 9 and 10 show the results. wind water w1 M M l    (6) Fig. 9 Wind heeling lever, lw1, evaluated by the tests Fig. 10 Wind heeling lever, lw1, compared with the GZ curve Standard Criterion Wind test Wind + Drift tests Drift test 40302010 Angle of Heel (deg.) 0 −0.05 −0.025 0.025 0.05 0.075 0.1 0.125 0.15 0 −10−20 IW1[m] 10−10 −0.5 0.5 1 −1 −20 0 0 20 30 GZ GZ,Iw1[m] Standard Criterion Wind + Drift tests 40 50 Angle of Heel [deg]
  • 88. 74 In Fig. 9, the heeling levers due to wind ( wind M  ) and drift motion ( water M  ) 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 1r 0.7  (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  ). S. Ishida et al.
  • 89. Evaluation of the Weather Criterion 75 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:  1 1 90 r r rs N     (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. S=1/60 =1/40 =1/26.1 0 1 2 3 4 5 6 7 Wave frequency / Natural roll frequency RollAmplitude/Waveslope
  • 90. 76 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 S. Ishida et al.
  • 91. 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.
  • 92. 78 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 S. Ishida et al.
  • 93. M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, 79 Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_5, © Springer Science+Business Media B.V. 2011 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 well- studied 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:
  • 94. Y. Nechaev80 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
  • 95. 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 on- board 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
  • 96. Y. Nechaev82 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:
  • 97. Evolution of Analysis and Standardization of Ship Stability 83 ],[,0),( ,)(),,,( 0 00 TtttXF XtxtYXf dt dx   (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; ),,( tYXf is a vector-valued function describing the dynamical system and 00 )( Xtx  are initial conditions. The solution of a problem (1) results in allowable values of output, used for formulation of stability criteria: . ,z| ,z| ;,,1, min ** 1 * 11 ** 1 * 111 1 KGKG zFGQ zFGQ JjQR CR jJ j j     … (2) Here J ,,1 … are the area of possible values of criteria, taking into account uncertainty and incompleteness of the initial information; CRKG — 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      kkCRG dCdCCCPZZP ,,,, 11 …… (3) where  is the area where variables kCC ,,1 … satisfy the following inequality CRkG ZCCZ ),,( 1 … (4) An alternative formulation of a problem is that the value of CRKG can be exceeded with the given probability 0P : 0)( PZZR CRG  (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
  • 98. Y. Nechaev84 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 ),,1(),( niaFi … 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 ),,1(, kja j … , or a function of the solution of differential equations. The coefficients of mathematical model ja , satisfying given parametrical, functional and criteria restrictions, make an allowable area aE in a space of criteria )( aEF . Setting the accuracy of approximation of parameters ja as ),,1(, kjj … , and the accuracy of criteria
  • 99. Evolution of Analysis and Standardization of Ship Stability 85 ),,1(, nii … , 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:  E n P n EP FFFFF  ,,11 … (6) where index P stands for calculation and E for experiment. The approximation of area aE is necessary for the evaluation of coefficients of a mathematical model with the given accuracy. It is achieved by finding values such as: NraFaaFaF r ,,1),(),()(min … (7) under conditions for determining allowable area aE ** i E i P i FFF  (8) Here 1a parameters that are within the boundaries *** jjj aaa  , N – number of experiments; ** iF - 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: ),,( 1 iiimi VSWS  , (9) where iS is the modeled state of a ship in the i-th instant of time; while 1iS is the state corresponding to the previous instant of time i – 1; iW and iV 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 mCcSY  , (10) where S is a domain limiting meaningful behavior of model; mC 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:
  • 100. Y. Nechaev86          SYSY SY SY if, if0 ,  . (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): ),,,(),,( ];),(),([5.0),(],),(),([5.0),( )];cos(),(),([),,( minmaxminmax tlDtM llllll tllDtMM kW kW       (12) where max),( l and min),( l 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
  • 101. 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. Fig. 2 Nonlinear function, describe of righting component: 1 – initial function; 2 – transform function; 3 – wave shape The geometric interpretation of function ),,( tM  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 Monte- Carlo, method of the moments, method of action functional, equation of Fokker- Plank-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 o     0  o 0 80 20 60 40 0,3 0,2 0,1 0 -0,1 M(,,t)/D 1 2 3 k  o 0 30 60 90 0 60 20 80 40 0.3 0.2 0.1 0 -0.1 1 2 M(,,t)/D o     0 o 0 80 20 60 40 0,2 0 -0,2 1 23M(,,t)/D k  o 0 30 60 90 0 60 20 80 40 0.3 0.2 0.1 0 -0.1  1 2 M(,,t)/D
  • 102. Y. Nechaev88 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 ).sgn)]}((cos[)(1{ ),,( 22 0    attkt JJ tM xx   (13) Parameter (t) and phase (t) are considered random variables with normal and uniform distribution, respectively. ]2,0[)(,})]([)],([{)( 0 **   ttDtMt (14) The hypothesis of Gaussian distribution of the process (t) is checked by the analysis of the time history of process. The correlation function )(* tR of the process (t) is estimated using the data of real sea waves. Calculations show, that the function )(* tR can be presented with the following approximation: ),)((cos|]|)(exp[)]([)( **   tttDR (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 )(* tD  and autocorrelation function )(* R . The filter is described by system of the differential equations: ),()(2)()()()( );()(2)()()()( 2 * 1 * tWDtt td d tWDtt td d       (16) where .0(0)(0)[;,(0)[ ;)0([,0)([);()()([ ;0)()([);()()([ ** 2 * * 2 *** 22 * 21 ** 11 *       MDDDM DMtMtWtWM WtWMtWtWM (17)
  • 103. Evolution of Analysis and Standardization of Ship Stability 89 Here )(1 tW and )(2 tW 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 saddle- type 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 way . 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 cycle- attractor can lose stability through several scenarios. A passing group of large s
  • 104. Y. Nechaev90 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 small- amplitude 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 radio- physics, 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
  • 105. 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). 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:                       1 cos )( )( )( )( 32 3 1 2 3 2 1 xBxkx x xf xf xf xf (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) 0 3 3 2 2 1 1          k x f x f x f (19) The linearized system leads to the following matrix            000 sin3 010 3 2 1 xBkxA (20) t 1 2 A B
  • 106. Y. Nechaev92 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 ),,1( mz … . 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
  • 107. 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. ),,1(00 mjNqN jj … (21) where 0 N is the predicted total number of extreme situations for the considered period of time; jq is the fraction of situations for the j-th class in the distribution F(w), determined by the formula )( maxmin jjj wwwPq  (22) The value jq 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:      2/120200 1 ][][][ jj qNNqN   (23) random deviation  20 ][Nqj and  20 ][ jqN  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 pre- processed and represents a time-series reconstructed on the basis of the Takens’ theorem. According to the Takens’ theorem, a phase portrait restored as: )](,),([])1([ ],),(),([)]([)( 1 tztzmty tytytytZ m… …     (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 ),( 21 tt . Responses of the dynamical system corresponding to all possible initial conditions are compared.
  • 108. Y. Nechaev94 If the interval of time )( 12 tt  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):           DxRC SDXP t nr T Ttx    100 0 2 ,,, minlnlim    (25) It allows estimating probability xP 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 . 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 уз
  • 109. 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: G NN k k NNN t N                           (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 dsnlin GGGG  (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.
  • 110. Y. Nechaev96 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     i iCR i x NNKG KGKG yd yy       1 , 1 y;100 0,0 2 exp 2 1 2 0 2 2                         (27) where KG, CRKG are the actual and critical position of center of gravity. А B C D E F
  • 111. 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 zero- probability. Also whether it is real when the conjunction of events having zero- probability 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 10 P . 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  Ni /*  of a random variable  is an unbiased estimate 0)( * E and converges   1)|(| * P 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).
  • 112. Y. Nechaev98 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 ];[ 00   XX . Then according to the requirements when 0 * XX  , the norm is observed exactly, while in reality 0X . The area of acceptance of a hypothesis 0XX  , i.e. the interval (  XX ,1 ) equals  ( - significance value). Then the probability of a type II error for deviation from hypothetical value qX 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. Xmin Xmax Fig. 9 Errors of practical use of stability criteria: 0X – parameter; ),( minmax XX - 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.
  • 113. 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. 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 5.5.5.
  • 114. Y. Nechaev100 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. 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 Belenky V, de Kat J, Umeda N (2008) Toward Marine Tech 45(2):122–123 performance-based criteria for intact stability.
  • 115. 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
  • 116. Y. Nechaev102 Nechaev Y (2002) Principle of competition at neural network technologies realization in on- board 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
  • 117. M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, 103 Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_6, © Springer Science+Business Media B.V. 2011 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 that 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, initiating rule aking 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 are rules ?m
  • 118. D. Vassalos and A. Jasionowski104 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
  • 119. SOLAS 2009 – Raising the Alarm 105 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 j I i jiij RAspwA 1 1 ; (1) Where: A/R Attained/Required Index of Subdivision j loading condition (draught) under consideration 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 Draught and GM
  • 120. D. Vassalos and A. Jasionowski106 I set of all feasible flooding scenarios comprising single compartments or groups of adjacent compartments for each j wj probability mass function of the loading conditions (draughts) pi probability mass function of the extent of flooding (that the compartments under consideration are flooded) sij 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: )(sEA  (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 collision- related 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”. 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 3 Probability of Survival – “S-Factor”
  • 121. 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. Fig. 1 The SEM Methodology θ critical GZmax G f h
  • 122. D. Vassalos and A. Jasionowski108 Accounting finally for the likelihood of occurrence of this sea state at the time of casualty led to the following relationship (Figure 2): 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. Fig. 3 Regression of GZ range s = exp(-exp (f –8h +1.3)s = exp(-exp (f –8h +1.3) Range vs. Hs 0 1 2 3 4 5 6 7 8 9 10 0 5 10 15 20 25 30 35 40 45 Range (deg.) Hscritical(m)
  • 123. SOLAS 2009 – Raising the Alarm 109 Fig. 4 Regression of GZ area Fig. 5 Regression of GZ max 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 Ro- Ro vessels, yielded the following final formulation (Figure 6): Area vs. Hs 0 1 2 3 4 5 6 7 8 9 10 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 GZ Area (m-rad) Hscritical(m) GZmax vs. Hs 0 1 2 3 4 5 6 7 8 9 10 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 GZmax (m) Hscritical(m)
  • 124. D. Vassalos and A. Jasionowski110 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). 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. s = [ (GZmax / 0.12) (Range / 16) ]1/4s = [ (GZmax / 0.12) (Range / 16) ]1/4 0 1 2 3 4 5 6 7 8 9 10 0 GZmax [m] Hscrit[m] 0 0.5 1 sfactor Conventional RoRo Cruise PCLS01 midship damage 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
  • 125. SOLAS 2009 – Raising the Alarm 111  Examining also Figure 7, leads to some additional and even more disturbing observations:  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. 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. All PASS ships 0,4 0,5 0,6 0,7 0,8 0,9 1 1,1 0 25 50 75 100 125 150 175 200 225 250 275 300 Ls (m) Rnew Rnew (SLF) Rnew (HARDER) Linear (Rnew (SLF)) Linear (Rnew (HARDER)) The single result on the cruise ship, which was never used, shows that even
  • 126. D. Vassalos and A. Jasionowski112 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 (MF 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 post- HARDER). S? effect), a
  • 127. SOLAS 2009 – Raising the Alarm 113 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. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 C1 - STD Opened A=0.638 0 0.005 0.01 0.015 0.02 0.025 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 C1 - STD Opened A=0.638
  • 128. D. Vassalos and A. Jasionowski114 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 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 time- domain 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: 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),
  • 129. 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-CT- 1998-00028. 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.
  • 130. 2 Stability of the Intact Ship
  • 131. M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, 119 Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_7, © Springer Science+Business Media B.V. 2011 Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas A. Yucel Odabasi and Erdem U 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 depende 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 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 çer nt on initial bias it causes amplitude of rolling motion to take large values after a few cycles. Thus,
  • 132. A.Y. Odabasi and E. U er120 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. tCosKxxxxx  ~ ˆ2 3 3 2 2 2 0  (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. tSinF)x1()x1(xxx  (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 ç 
  • 133. 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).       windR MtEMDI   , (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    tCosEtE w  where Ew is the amplitude and Ω the angular frequency Mwind : Wind Moment
  • 134. A.Y. Odabasi and E. Uçer122 In this study, for the sake of simplicity, the following approximations have been used  3 21)(  kkM R  and  sD ),( where Δ is the displacement and s is the linear damping coefficient Under the above assumptions Equation (3) can be written as   windw MtCosEkksI  3 21  (4) Dividing both sides of the equation by the mass moment of inertia, the expression given in Equation (5) is obtained. ruzFtCosFcc  3 21ˆ  (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 tCosFxdxdxxx  3 2 2 1 2 0ˆ  (6) where 2 s21 2 0 c3c  , 2s1 c3d  , 22 cd  , 3 s2s1ruz ccF  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.
  • 135. 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 d2 ˆ F 0.005 0.610257 0.001975 0.100 0.607020 0.039499 0.200 0.597179 0.078998 0.131664 0.04,0.07,0.10,0.13 0.08,0.10,0.12 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. tT n n  For n=0,1,2…… where ε is a small parameter Time derivatives then can be re-written as ................ 10 1 1 0 0        DD Ttd Td Ttd Td dt d  (7a)   ..........22 20 2 1 2 10 2 02 2  DDDDDD td d  (7b) 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;        100100 2 322 0 TTiTTi ee f xgxdxxx     (8) If the expressions shown in Equation (7a) and (7b) are substituted into Equation (8), Equation (9) is obtained.
  • 136. A.Y. Odabasi and E. Uçer124       ...... 2 22 10010032 1 2 0 2 0 20 22 1 2 10 2 0     TTiTTi ee f xgxd xDxDx xDDxDxDDxD     (9) The solution of above equation can be represented by an expansion as ...........,....),,( ,....),,(,....),,( 2102 2 21012100   TTTx TTTxTTTxx   (10) 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: 0 2 00 2 0 0  xxD  (11a)  100100 2 2: 3 0 2 0000101 2 01 2 0 1 TiTiTiTi eeee f xgxdxDxDDxxD      (11b) 1 2 010 01100 2 1 0201102 2 02 2 0 2 32 22: xgxxxd xDxDxD xDDxDDxxD      (11c) The solution of Equation (11a) is 0000 )()( 110 TiTi eTAeTAx    (12) where )( 11 1 )( 2 1 )( Ti eTaTA   and )( 11 1 )( 2 1 )( Ti eTaTA   , A and A are the 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 00Ti e  and 00Ti e  to zero.
  • 137. Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 125   0 2 32 12 010  Ti e f gAAAiADi   (13a)   0 2 32 12 010   Ti e f gAAAiADi   (13b) 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.   0 22 1 0     TSin fa a (14a)   0 28 3 1 0 3 0     TCos f a g a (14b) where (΄) symbol represents derivative taken with respect to T1. Defining  1T and  , the above equation becomes     Sin fa a 022  (15) The above equations can also be written in the form of the equations below.             Sin fa a 022 (16a)            Cos f a g aa 0 3 0 28 3 (16b) The first order approximate solution of biased roll equation is written as:      tCosax (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).
  • 138. A.Y. Odabasi and E. Uçer126 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. Fig. 1 a-σ graph The effect of initial bias angle 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. 0.6 m=0.04 F=0.12 d2 =0.131664 0.5 0.4 0.3 0.2 0.1 0 -0.15 -0.12 -0.09 -0.06 -0.03 0 a 0.03 0.06 0.09 0.12 0.15 s Bias=0.20 rad. Bias=0.15 rad. Bias=0.10 rad. 0.6 w0=0.597179 d1=0.078998 d2=0.131664 m=0.04 0.5 0.4 0.3 0.2 0.1 0 -0.15 -0.12 -0.09 -0.06 -0.03 0 a 0.03 0.06 0.09 0.12 0.15 s F=0.12 F=0.1 F=0.08
  • 139. 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, 02  , 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, ω0 2 =Ω2 +εΔ, linear damping coefficient, quadratic and cubic term of restoring
  • 140. 128 moment and finally amplitude of excitation force have been assumed to be order of ε. Under these assumptions biased roll equation were written as  xtCosfxgxdxxx  322  (18) The above equations can be written in more efficient form as            xSinSinfCosCosf xgxdx xx    32 2 (19) where ψ = Ω t+β The solution is assumed to have the form (Bogoliubov and Mitropolsky 1961),   ...........,,1   tauCosax (20) and a and β in the above equation must satisfy the following expressions .........),(),( 2 2 1   aAaA dt da (21a) ..........),(),( 2 2 1    aBaB dt d (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.   cos22 11 2 aCosBaSinAxx  (22)                      SinSinfCosCosf ag Cos ag Cos ad ad SinaCosa xSinSinfCosCosfxgxdx 4 3 3 4 2 2 2 3 32 2 32 (23) A.Y. Odabasi and E. U erç
  • 141. 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:  SinfaA  12 (24a) Cosf ag aBa  4 3 2 3 1 (24b)  3 32 2 62 2 3 2 2 2 2 1 Cos ag Cos adad u       (25) If A1 and B1 are substituted into the Equation (21a) and Equation (21b), the following expressions are obtained:   Sin Fa td ad   22 ˆ (26a)   Cos a Fad td d        28 3 2 2 2 22 0 (26b) The first order approximate solution of biased roll equation is the given as:      tCosax (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), Equation (28) is used. Equation (28) is obtained by making tCos F yx    22 0 transformation in the Equation (6).     03 2 2 0             tCosyg tCosydtSiny yy   (28) where 22 0    F
  • 142. 130 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 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. Fig. 6 Comparison of second order approximate MSM and numeric solution when 0 1.5 ω=0.597179 d=0.078998 σ=0.1 F=0.12 1 0.5 0 0 50 t x(t) 100 150 200 250 300 350 400 -0.5 -1 -1.5 MSM O(ε^2) Numeric Fig. 12-17. As can be seen from the figures, the second order approximate A.Y. Odabasi and E. U erç
  • 143. Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 131 Fig. 7 Comparison of second order approximate MSM and numeric solution when 02 Fig. 8 Comparison of second order approximate MSM and numeric solution when 20 Fig. 9 Comparison of first order approximate MSM and numeric solution when 0 0 x(t) -0.50 0.00 -1.00 0.50 1.00 MSM O(ε^2)x(0)=0 dx/dt=0.56 ω=0.59718 Ω=1.29436 d=0.078998 1.50 50 100 150 200 Numeric 250 300 350 400 0 50 100 150 200 250 300 Numeric MSM O(ε^2) 350 400 0 0.2 x(t) -0.2 0.4 -0.4 0.6 -0.6 1 x(0)=0.08045 dx/dt=0.21482 ω=0.59718 Ω=0.28359 d=0.078998 0.8 -0.8 -1 -1.2 t 0 x(t) 1 0.5 1.5 0 -1.5 -0.5 -1 50 100 x(0)=0 dx/dt=0.459 ω=0.59718 Ω=0.69718 d=0.078998 150 200 t 250 300 Numeric MSM O(ε) 350 400
  • 144. 132 Fig. 10 Comparison of first order approximate MSM and numeric solution when 02 Fig. 11 Comparison of first order approximate MSM and numeric solution when 20 Fig. 12 Comparison of second order approximate asymptotic and numeric solution when 0 0 -0.5 -1 0.5x(t) 1 1.5 0 50 100 150 x(0)=0 dx/dt=0.56 ω=0.59718 Ω=1.29436 d=0.078998 200 t 250 MSM O(ε) Numeric 300 350 400 t -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 x(t) 0.2 0.4 0.6 x(0)=0.1 dx/dt=0.2 ω=0.59718 Ω=0.2835 d=0.078998 0.8 1 0 50 100 150 200 250 300 350 Numeric MSM O(ε) 400 t 0 50 100 150 200 250 300 350 400 0 x(t) -0.5 -1 1 Numeric Asymptotic O(ε^2) μ=0.1,F=0.12, Ω=0.697179, ω=0.597179 x(0)=0 dx/dt=0.535 0.5 1.5 A.Y. Odabasi and E. U erç
  • 145. Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 133 Fig. 13 Comparison of second order approximate asymptotic and numeric solution when 02 Fig. 14 Comparison of second order approximate asymptotic and numeric solution when 20 Fig. 15 Comparison of first order approximate asymptotic and numeric solution when 0 t 0 x(t) 1 -1 0.5 -0.5 1.5 0 50 100 μ=0.04,F=0.12, Ω=1.294358,ω=0.597179 x(0)=0 dx/dt=0.56 Bias=0.2 rad. 150 200 250 300 350 Asymptotic O(ε^2) Numeric 400 0 x(t) 1 -1 0.5 -0.5 -1.5 1.5 0 50 100 Ω=0.285755 F=0.12 x(0)=0.25 y(0)=0.268 Bias=0.2 rad. 150 200 250 300 350 400 t Asymptotic O(ε^2) Numeric 0 50 100 150 μ=0.1,F=0.12, Ω=0.697179,ω=0.597179 x(0)=0 dx/dt=0.535 200 250 300 350 400 0 1 -1 0.5 -0.5 -1.5 1.5 x(t) t Asymptotic O(ε) Numeric
  • 146. 134 Fig. 16 Comparison of first order approximate asymptotic and numeric solution when 02 Fig. 17 Comparison of first order approximate asymptotic and numeric solution when 20 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. 0 50 100 150 200 250 300 350 400 0 0.5 -0.5 -1 t 1.5 1 x(t) Asymptotic O(ε) Numeric x(0)=0 y(0)=0.56 ω=0.597179 Ω=1.29436 Bias=0.2 rad. 1.2 1 0.8 0.6 0.4 0.2 0 0 50 100 t 150 200 250 300 Asymptotic O(ε) Numeric 350 400 x(t) -0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4 ω=0.597179 Ω=0.285755 Bias=0.2 rad. A.Y. Odabasi and E. U erç
  • 147. 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 Wave excitation Initial Bias Angle θs=0.2 rad. Initial Bias Angle θs=0.1 rad. Initial Bias Angle θs=0.005 rad. |F|=0.12 - - - |F|=0.10 - - - |F|=0.08 - Stable Bound for µ ≥0.13 Stable Bound for µ ≥0.1 Table 3 Stable bounds in main resonance region for σ = 0.1 Amplitude of Wave excitation Initial Bias Angle θs=0.2 rad. Initial Bias Angle θs=0.1 rad. Initial Bias Angle θs=0.005 rad. F=±0.12 Stable Bound For µ > 0.10 Stable Bound For µ ≥ 0.07 Stable Bound For µ ≥ 0.04 F=±0.10 Stable Bound For µ > 0.07 Stable Bound For µ > 0.04 Stable Bound For µ ≥ 0.04 F=±0.08 Stable Bound For µ > 0.04 Stable Bound for µ ≥0.04 Stable Bound 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, 0)0(x  and 634.0)0(x  . 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).
  • 148. 136 Table 5 Some points out of the stable bound tt5.24 t (sn.) x (rad.) x (rad/s) 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 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). 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, 1.8 1.6 1.4 1.2 0.8 1 2 3 4 x y 0.6 Fig. 20. A.Y. Odabasi and E. U erç
  • 149. Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 137 Fig. 20 Solution orbit in x- x phase space for initial condition x(19.5)=0.19428 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.
  • 150. 138 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 A.Y. Odabasi and E. U erç
  • 151. 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 20 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.
  • 152. 140 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 A.Y. Odabasi and E. U erç
  • 153. M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, 141 Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_8, © Springer Science+Business Media B.V. 2011 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
  • 154. H. Nowacki and L.D. Ferreiro142 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?
  • 155. 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 preserve