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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: http://www.springer.com/series/5980
- 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 (www.springer.com) 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 Jean.Dekat@maersk.com masn@peno.coppe.ufrj.br Vadim L. Belenky Naval Surface Warfare Center Carderock D 20817-5700 West Bethesda Maryland USA vadim_belenky@hotmail.com Kostas Spyrou School of Naval Architecture National Technical University of Athens Iroon Polytechnion, 157 73 Athens Greece and Marine k.spyrou@central.ntua.gr 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 umeda@naoe.eng.osaka-u.ac.jp 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 vadim_belenky@hotmail.com A Basis for Developing a Rational Alternative to the Weather Criterion: Problems and Capabilities K.J. Spyrou spyrou@deslab.ntua.gr Conceptualising Risk Dracos Vassalos d.vassalos@strath.ac.uk Evaluation of the Weather Criterion by Experiments and its Effect to the Design of a RoPax Ferry Shigesuke Ishida ishida@nmri.go.jp Evolution of Analysis and Standardization of Ship Stability: Problems and Perspectives Yury Nechaev int@csa.ru SOLAS 2009 – Raising the Alarm Dracos Vassalos d.vassalos@strath.ac.uk 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 nowacki@naoe.tu-berlin.de xiii E. Uçer ucerer@itu.edu.tr
- 12. List of Corresponding Authorsxiv The Effect of Coupled Heave/Heave Velocity or Sway/Sway Velocity Initial Conditions on Capsize Modeling Leigh S. McCue mccue@vt.edu The Use of Energy Build Up to Identify the Most Critical Heeling Axis Direction for Stability Calculations for Floating Offshore Structures Joost van Santen joost.vansanten@gustomsc.com Some Remarks on Theoretical Modelling of Intact Stability Naoya Umeda umeda@naoe.eng.osaka-u.ac.jp 3 Parametric Rolling An Investigation of Head-Sea Parametric Rolling for Two Fishing Vessels Marcelo de Almeida Santos Neves masn@peno.coppe.ufrj.br Simple Analytical Criteria for Parametric Rolling K.J. Spyrou spyrou@deslab.ntua.gr Experimental Study on Parametric Roll of a Post-Panamax Containership in Short-Crested Irregular Waves Hirotada Hashimoto h_hashi@naoe.eng.osaka-u.ac.jp Model Experiment on Parametric Rolling of a Post-Panamax Containership in Head Waves Harukuni Taguchi taguchi@nmri.go.jp Numerical Procedures and Practical Experience of Assessment of Parametric Roll of Container Carriers Vadim Belenky vadim_belenky@hotmail.com Parametric Roll and Ship Design Riaan van’t Veer riaan.vantveer@sbmoffshore.com
- 13. List of Corresponding Authors xv Parametric Roll Resonance of a Large Passenger Ship in Dead Ship Condition in All Heading Angles Toru Katayama katayama@marine.osakafu-u.ac.jp Parametric Rolling of Ships – Then and Now J. Randolph Paulling RPaulling@aol.com 4 Broaching-to Parallels of Yaw and Roll Dynamics of Ships in Astern Seas and the Effect of Nonlinear Surging K.J. Spyrou spyrou@deslab.ntua.gr Model Experiment on Heel-Induced Hydrodynamic Forces in Waves for Realising Quantitative Prediction of Broaching Hirotada Hashimoto h_hashi@naoe.eng.osaka-u.ac.jp Perceptions of Broaching-To: Discovering the Past K.J. Spyrou spyrou@deslab.ntua.gr 5 Nonlinear Dynamics and Ship Capsizing Use of Lyapunov Exponents to Predict Chaotic Vessel Motions Leigh S. McCue mccue@vt.edu Applications of Finite-Time Lyapunov Exponents to the Study of Capsize in Beam Seas Leigh S. McCue mccue@vt.edu Nonlinear Dynamics on Parametric Rolling of Ships in Head Seas Marcelo A.S. Neves, masn@peno.coppe.ufrj.br 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 ikeda@marine.osakafu-u.ac.jp
- 14. List of Corresponding Authorsxvi A Study on the Characteristics of Roll Damping of Multi-hull Vessels Toru Katayama katayama@marine.osakafu-u.ac.jp 7 Probabilistic Assessment of Ship Capsize Capsize Probability Analysis for a Small Container Vessel E.F.G. van Daalen E.F.G.van.Daalen@marin.nl Efficient Probabilistic Assessment of Intact Stability K.J. Spyrou spyrou@deslab.ntua.gr Probability of Capsizing in Beam Seas with Piecewise Linear Stochastic GZ Curve Vadim Belenky vadim_belenky@hotmail.com Probabilistic Analysis of Roll Parametric Resonance in Head Seas Vadim L. Belenky vadim_belenky@hotmail.com 8 Environmental Modeling Sea Spectra Revisited Maciej Pawłowski mpawlow@pg.gda.pl On Self-Repeating Effect in Reconstruction of Irregular Waves Vadim Belenky vadim_belenky@hotmail.com New Approach To Wave Weather Scenarios Modeling Alexander B. Degtyarev deg@csa.ru 9 Damaged Ship Stability Effect of Decks on Survivability of Ro–Ro Vessels Maciej Pawłowski mpawlow@pg.gda.pl
- 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 ikeda@marine.osakafu-u.ac.jp Exploring the Influence of Different Arrangements of Semi-Watertight Spaces on Survivability of a Damaged Large Passenger Ship Riaan van’t Veer riaan.vantveer@sbmoffshore.com Time-Based Survival Criteria for Passenger Ro-Ro Vessels Dracos Vassalos d.vassalos@strath.ac.uk Pressure-Correction Method and Its Applications for Time-Domain Flooding Simulation Pekka Ruponen pekka.ruponen@napa.fi 10 CFD Applications to Ship Stability Applications of 3D Parallel SPH for Sloshing and Flooding Dracos Vassalos d.vassalos@strath.ac.uk Simulation of Wave Effect on Ship Hydrodynamics by RANSE Dracos Vassalos d.vassalos@strath.ac.uk A Combined Experimental and SPH Approach to Sloshing and Ship Roll Motions Luis Pérez-Rojas luis.perezrojas@upm.es 11 Design for Safety Design for Safety with Minimum Life-Cycle Cost Dracos Vassalos d.vassalos@strath.ac.uk
- 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 AJPETERS@qinetiq.com A Naval Perspective on Ship Stability Arthur M. Reed arthur.reed@navy.mil 13 Accident Investigations New Insights on the Sinking of MV Estonia Dracos Vassalos d.vassalos@strath.ac.uk Experimental Investigation on Capsizing and Sinking of a Cruising Yacht in Wind Naoya Umeda umeda@naoe.eng.osaka-u.ac.jp
- 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(2xG/ 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
- 35. References ABS (2004) American bureau of shipping guide for the assessment of parametric roll resonance in the design of container carriers. Houston, Texas Ananiev D M (1966) On surf-riding in following seas. Trans of Krylov Soc 73. Russian Ananiev D M, Loseva L (1994) Vessel’s heeling and stability in the regime of manoeuvring and broaching in following seas. Proc. 5th Intl Conf on Stab of Ships and Ocean Veh. Melbourne, Florida Belenky VL (1993) A capsizing probability computation method. J. Ship Res 37:200-207 Belenky VL, Sevastianov NB (2007) Stability and safety of ships - risk of capsizing (second edition). The Soc of Nav Archit and Mar Eng (SNAME). Jersey City, NJ Belenky V, de Kat JO, Umeda N (2008) Toward performance-based criteria for intact stability. Mar Technol 45:101-123 Blume P (1979) Experimentelle bestimmung von koeffizienten der wirksamen rolldaempfung und ihre anwendung zur abschaetzung extremer rollwinkel. Schiffstechnik, 26 Blume P (1987) Development of new stability criteria for modern dry cargo vessels. Proc. 5th Intl. Symp. on Pract Des of Ships and Other Float Struct. Trondheim, Norway Boroday IK (1967) Statistical characteristics of stability and the probability of capsize of a ship running on any course in irregular seas. Doc of the USSR Expert of the IMCO, Working Gr on Stab of Fish Vessels, IMO. London Boroday IK (1968) Statistical characteristics and probability of capsizing of ship heading with arbitrary course in irregular seas. Trans. of russian register of shipping— theoretical and practical problems of stability and survivability. Transp Publ. Moscow, Russian Boroday IK, Netsvetaev YA, (1969) Ship motion in a seaway. Sudostroenie. Leningrad, Russian Brunswig J, Pereira R, Kim D (2006) Validation of parametric roll motion predictions for a modern ship containership design Proc. 9th Intl. Conf. on the Stab of Ships and Ocean Veh. Rio de Janeiro, Brazil 157-168 Bulian G (2006) Development of analytical nonlinear models for parametric roll and hydrostatic restoring variations in regular and irregular waves. PhD Thesis. Univ of Trieste Bulian G, Francescutto A (2006) On the effect of stochastic variations of restoring moment in long-crested irregular longitudinal sea. Proc 9th Int Conf. on Stab of Ships and Ocean Veh. Rio de Janeiro, Brazil. 1:131-146 Bulian G, Francescutto, A Lugni C (2004) On the nonlinear modeling of parametric rolling in regular and irregular waves. Int Shipbuild Prog. 51 2/3:173-203 Bulian G, Francescutto, A Lugni C (2006) Theoretical, numerical and experimental study on the problem of ergodicity and ‘practical ergodicity’ with an application to parametric roll in longitudinal long crested irregular sea. Ocean Eng, 33:1007-1043 Bulian G, Francescutto A, Maccari A (2008) Possible simplified mathematical models for roll motion in the development of performance-based intact stability criteria– extended and revised version. University of Trieste Burcher RK (1990) Experiments into the capsize of ships in head seas. Proc 4th Int Conf on Stab of Ships and Ocean Veh, Naples, 82-89 Davidson KSM (1948) A note on the steering of ships in following seas. Proc 7th Int Congr of Appl Mech. London, 554-556 DeKat JO, Thomas WL (1998) Extreme rolling, broaching and capsizing— model tests for validation of numerical ship motion predictions. Proc 22nd Symp on Nav Hydrodyn. Washington, DC DuCane P, Goodrich GJ (1962) The following sea, broaching and surging. Trans. RINA 104 Dunwoody AB (1989a) Roll of a ship in Astern Seas—metacentric height spectra. J Ship Res, 33 3:221-228 Dunwoody AB (1989b) Roll of a ship in Astern Seas –response to GM fluctuations. J Ship Res, 33 4:284-290 C.C. Bassler et al.20
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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. http://www.mariecurie.org/annals/volume1/spyrou.pdf Spyrou KJ (2000) On the parametric rolling of ships in a following sea under simultaneous nonlinear periodic surging. Phil Trans R Soc London, A 358:1813-1834 Spyrou KJ (2005) Design criteria for parametric rolling. Ocean Eng Int, Canada 9 1:11-27 Spyrou KJ (2006) Asymmetric surging of ships in following seas and its repercussions for safety. Nonlinear Dyn, 43:149-172 Spyrou KJ, Tigkas I, Scanferla G, Themelis N (2008) Problems and capabilities in the assessment of parametric rolling. Proc 10th Int Ship Stab Workshop. Daejeon, Korea 47-55 Sverdrup HU, Munk WH (1947) Wind, sea and swell, theory of relations for forecasting. Hydrogr Off Publ 601 Themelis N, Spyrou KJ (2007) Probabilistic assessment of ship stability trans. SNAME, 115:181-206 Thomas G, Renilson M (1992) Surf-riding and loss of control of fishing vessels in severe following seas. Trans RINA 134:21-32 Umeda N (1990) Probabilistic study on surf-riding of a ship in irregular following seas. Proc of 4th Int Conf onn Stab of Ships and Ocean Veh. Naples 336-343 Umeda N, Renilson MR (1992) Broaching— A dynamic analysis of yaw behavior of a vessel in a following sea. In Wilson PA (ed) Maneuvering and Control of Mar Craft, Comput Mech Publ. Southampton 533-543 Umeda N, Yamakoshi Y (1993) Probability of ship capsizing due to pure loss of stability in quartering seas. Nav Archit and Ocean Eng: Sel Pap Soc of Naval Arch of Japan 30:73-85 Umeda N, Shuto M, Maki A (2007) Theoretical prediction of broaching probability for a ship in irregular astern seas. Proc 9th Int Ship Stab Workshop. Hamburg Umeda N, Hori M, Hashimoto H (2007) Theoretical prediction of broaching in the light of local and global bifurcation analysis. Int Shipbuild Prog 54 4:269-281 Vermeer H (1990) Loss of stability of ships in following waves in relation to their design characteristics. Proc of 4th Int Conf on Stab of Ships and Ocean Veh. Naples Vidic-Perunovic J, Jensen JJ (2009) Estimation of parametric rolling of ships – comparison of different probabilistic methods. Proc 2nd Int Conf on Mar Struct. Lisbon, Portugal Wahab R, Swaan WA (1964) Course-keeping and broaching of ships in following seas. J Ship Res 7 4
- 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 Bas

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