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[Marcelo almeida santos_neves,_vadim_l._belenky,_j(book_fi.org) [Marcelo almeida santos_neves,_vadim_l._belenky,_j(book_fi.org) Presentation Transcript

  • Contemporary Ideas on Ship Stability and Capsizing in Waves
  • 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
  • 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
  • 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.
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 1 Stability Criteria
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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).
  • 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
  • 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 &
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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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  • 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
  • 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
  • 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
  • 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
  • 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
  • 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:
  • 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
  • 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
  • 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
  • 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)
  • 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
  • A Basis for Developing a Rational Alternative to the Weather Criterion 35 sudden reduction of basin area beyond some critical wave forcing for all the members of this family i.e. that the concept is “robust” with respect to the characteristics of the restoring function. Fig. 6 Scaled “quintic” restoring curves In Fig. 7 are shown some typical results of this investigation for an initially hardening GZ at four different levels of damping. The points on the curves are identified with measurement of the basin area and scaling against the area of the corresponding unforced system. The curves confirm that for the fifth-order levers the concept of engineering integrity is applicable without any problem i.e. there is indeed a quick fall of the curve beyond some critical forcing. This critical forcing should be approximately equal to the forcing determined from the Melnikov method. An advantage however of the direct basin plot is that it allows to determine fractional integrities also like 90% 80% etc. as function of damping and excitation (90% integrity means that the remaining basin area is 90% of the initial). This is quite important for criteria development since it provides the necessary flexibility for setting the boundary line; for example one could base the maximum sustainable wave slope at 90% integrity rather than at 100% which might be too stringent.
  • 36 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
  • 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
  • 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
  • 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
  • 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
  • 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.
  • 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
  • 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.
  • 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
  • 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:
  • 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
  • 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
  • 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.
  • 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.
  • 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?
  • 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.
  • 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.
  • Conceptualising Risk 53 In the case of “an experiment” of operating a ship, however, whereby events involving fatalities can occur in the continuum of time, the pertinent sample space is customarily not adopted, and rather, the rate of occurrence of these events per unit time is used. The rate can be greater than unity, and obviously can no longer be referred to as probability. Thus, also use of the term “likelihood” would seem to be displaced here since it is synonymous with probability rather than frequency. Therefore, application of the term frequency per ship per year is customary, also since its quantification relies on the historical data for the relevant fleet. Lack of a well defined sample space, however, does not prevent one from using concepts of probability, such as e.g. distribution, expectation, etc, in analyses and analytical modelling. Thus, following from this preamble, it is proposed here that for the purpose of risk modelling the following assumptions are made: (a) The loss is assumed to be measured in terms of an integer number of potential fatalities among passengers that can occur as a result of activity relating to ship operation (b) The loss is a result of occurrence of a set of scenarios which can lead to this loss (c) Scenarios are intersections of a set of events (all must occur) and are identified by principal hazards2 , e.g. fire (d) The scenarios are disjoint (if one scenario occurs then the other does not) These assumptions are shown graphically in Fig 2 below. Fig. 2 Illustration of the concept of a compound event referred to as “fatality” by Venn’s diagram, (Jasionowski 2005), which is a union of mutually exclusive scenarios, whereby each scenario is an intersection of a set of relevant events and is identified by principal hazards, such as collision, fire, etc. 2 an event of a loss scenario is a hazard that materialised
  • A. Jasionowski and D. 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   
  • 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
  • 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).
  • 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)
  • 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  
  • 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
  • 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
  • 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.
  • 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, www.safedor.org Sames P (2005) Risk Based Design Framework: Design Decision Making. SAFEDOR, D5.1.2, Rev 7 Skjong R, Vanem E, Oyvind E (2005) Risk Evaluation Criteria. SAFEDOR, D4.5.2, DNV SLF 47/17 (2004) Sub-Committee On Stability And load Lines And On Fishing Vessels Safety Tagg R, Tuzcu C (2002) A Performance-based Assessment of the Survival of Damaged Ships – Final Outcome of the EU Research Project HARDER. Proc 6th Int Ship Stab Workshop, Webb Inst Vanem E, Skjong R (2004) Collision and Grounding of Passenger Ships – Risk Assessment and Emergency Evacuations. Proc 3rd Int Conf on Collision and Grounding of Ships, Izu Vassalos D (1999) Shaping ship safety: the face of the future. J. Marine technol36 (2):61-74 Vassalos D, Kim H, Christiansen G, Majumder J (2001) A Mesoscopic Model for Passenger Evacuation in a Virtual Ship-Sea Environment and Performance Based Evaluation. Pedestrian and Evacuation Dynamics, Duisburg Vassalos D, Jasionowski A, Guarin L (2005) Passenger Ship Safety – Science Paving the Way. Proc 8th Int Ship Stab Workshop, Istanbul Vrijling JK, Van Gelder PHAJM (1997) Societal Risk And The Concept Of Risk Aversion. Dep of Civil Eng, Delft Univ Of Technol, Delft www.safedor.org
  • 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
  • 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.
  • 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
  • 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.
  • 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
  • 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.
  • 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)
  • 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.
  • 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
  • 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.
  • 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]
  • 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.
  • 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 1.41.21.00.80.6 0 1 2 3 4 5 6 7 Wave frequency / Natural roll frequency RollAmplitude/Waveslope
  • 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.
  • Evaluation of the Weather Criterion 77 paper that 1  was enlarged by the small damping coefficient (N=0.011 at 20 degrees). On the other hand, the lw1 evaluated through all the combinations of the wind tests and drifting tests, complete procedures and simplified procedures, tends to make b/a larger than the standard criterion. It has to be noted that the leading cause of the fluctuations is the large variation in the vertical centre of hydrodynamic pressure when evaluated through the drifting tests. Fig. 12 shows the critical values of KG. From Table 2 and Fig. 12, it is recognized that the changes in the critical value of the vertical centre of gravity are more contained than b/a, but as much as 0.4m at the maximum from the standard criterion. Fig. 12 Variation of critical KG 7 Conclusions In this paper, the results of the alternative assessment of the weather criterion by model experiments are reported. Almost full tests, included in the interim guidelines, were conducted. Considerable fluctuations of the assessment and their reasons are clarified. In order to make the assessment more uniform and to remove the word of “interim” from the guidelines, more extensive confirmation from the experience gained through the application of the guidelines is needed. Acknowledments Some parts of this investigation were carried out as a research activity of the SPL research panel of the Japan Ship Technology Research Association in the fiscal year of 2005, funded by the Nippon Foundation. The authors express their sincere gratitude to the both organizations.
  • 78 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.
  • 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:
  • 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
  • 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
  • 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:
  • 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
  • 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
  • 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:
  • 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
  • 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
  • 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)
  • 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
  • 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
  • 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
  • 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
  • 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.
  • 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
  • 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.
  • 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
  • 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).
  • 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.
  • 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.
  • 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.
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  • 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
  • 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
  • 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
  • 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
  • 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”
  • 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
  • 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)
  • 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)
  • 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
  • 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
  • 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
  • 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
  • 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),
  • 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.
  • 2 Stability of the Intact Ship
  • 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,
  • 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 
  • 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
  • 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.
  • 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.
  • 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.
  • 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).
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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).
  • 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
  • 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.
  • 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
  • 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.
  • 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
  • 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
  • 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?
  • Historical Roots of the Theory of Hydrostatic Stability of Ships 143 To answer these questions it is not sufficient to look only at their final results and conclusions, but it is necessary to examine more closely the methodical approaches taken to the ship stability issue and to compare the trains of thought by which the individual authors arrived at their results. In this article we will review the developments that led to this historical stage when modern hydrostatic stability theory was founded. 2 Precursors 2.1 Archimedes ARCHIMEDES of Syracuse (ca. 287-212 B.C.), the eminent mathematician, mechanicist and engineering scientist in antiquity, is also the founder of ship hydrostatics and hydrostatic stability, which he established as scientific subjects on an axiomatic basis. ARCHIMEDES was brought up in the early Hellenistic era in the tradition of Greek philosophy, logical rigor and fundamental geometric thought. ARCHIMEDES was well educated in these subjects in Syracuse and very probably also spent an extended study period in Alexandria, the evolving Hellenistic center of science, at the Mouseion (founded in 286 B.C.). There he met many leading contemporary scientists, e.g., DOSITHEOS, ARISTARCHOS and ERATOSTHENES, with whom he maintained lifelong friendships and scientific correspondence. Steeped in this tradition of Greek geometry and mechanics, ARCHIMEDES learned how to excel in the art of deductive proofs from first premises usually based on conceptual models, i.e., models of thought, by which physical reality was idealized. But ARCHIMEDES was also unique, as many legends on his engineering accomplishments tell us, in applying practical observation to test his scientific hypotheses and to develop engineering applications, although these achievements are not mentioned in his own written work. We are fortunate that many, though not all, of ARCHIMEDES’ treatises are preserved, all derived from a few handwritten copies made in the Byzantine Empire during the 9th and 10th centuries and transmitted to posterity on circuitous routes to resurface essentially through the ARCHIMEDES revival during the Renaissance. Luckily, one Greek manuscript, which was later lost, had been translated into Latin by the Dominican monk Willem van MOERBEKE. This translation, which was published in 1269 and was later named Codex B, also contained ARCHIMEDES’ treatise “On Floating Bodies”, Books I and II. It became the only accessible reference to ARCHIMEDES’ work on hydrostatics for many centuries until in 1906 most surprisingly an old 10th century palimpsest was rediscovered in a Greek monastery in Constantinople by J. HEIBERG (Heiberg 1906-07). This palimpsest, called Codex C by HEIBERG, under the writing of a 12th century monk of a Greek prayerbook had retained significant traces of the
  • H. Nowacki and L.D. Ferreiro144 rinsed off text of a Greek manuscript from ARCHIMEDES including the Greek version of “On Floating Bodies”. ARCHIMEDES’ texts were soon reconstructed from this source, transcribed and translated into modern languages (Archimedes 2002). Meanwhile this palimpsest, which had disappeared in private possession in the aftermath of the Greek-Turkish war in 1920-22, has turned up again recently and is under new scientific evaluation at the Walters Art Museum in Baltimore (Netz and Noel 2007). It is on these sources primarily that we can base a reliable evaluation of ARCHIMEDES’ contributions to ship hydrostatics today. ARCHIMEDES preceded his work on hydrostatics by a number of other treatises establishing certain axioms of mechanics: 2.1.1 The Law of the Lever: In his treatise “The Equilibrium of Planes” ARCHIMEDES first deals with the equilibrium of moments about a fulcrum in a lever system. Although he claims to deduce this principle from geometric reasoning alone, it is actually understood today that the law of the lever is equivalent to the axiom of moment equilibrium in mechanics. Second, he introduces the concept of “centroids” of quantities (areas, volumes, weights) into which the quantities can be “lumped” as concentrated effects so that moment equilibrium is retained. Third, he proposes a method for finding the “compound centroid” of a system of components, e.g., a center of gravity. Finally, he proves the critical “centroid shift theorem” i.e., a rule for the shift of the system centroid when some quantity is added to, removed from or shifted within the system. All of these concepts and results are essential physical principles as prerequisites for his work on hydrostatics. 2.1.2 Quadrature: In his treatise “The Quadrature of the Parabola” ARCHIMEDES illustrates by the example of the parabola how the Greeks determined areas and volumes of elementary shapes without the availability of calculus. Here he uses a method, well-known since EUDOXUS (Boyer 1939, 1949), based on an inscribed polygonal approximant which is continually refined by interval halving until under the given premises it converges to the given curve within a specified error tolerance. This type of deduction was later called “method of exhaustion”. Thus the quadrature problem is reduced to that of evaluating the sum of a finite, truncated or sometimes even infinite series of approximations. This method of quadrature generally is not equivalent to calculus for lack of a limiting process to infinitesimal step width, but has nonetheless inspired many future developments until this day.
  • Historical Roots of the Theory of Hydrostatic Stability of Ships 145 2.1.3 The Method of Mechanical Theorems: This famous treatise, which was actually also discovered in the palimpsest of 1906, explains how ARCHIMEDES used a reasoning based on principles of mechanics (like moment equilibrium of volume quantities) in deriving geometrical results (like volume centroids). He regarded such findings as propositions for later rigorous deduction by strictly geometrical proofs. 2.1.4 On Floating Bodies: In this treatise ARCHIMEDES makes use of all these prior results and proceeds to lay the foundations for ship hydrostatics and stability in the following steps (Archimedes 2002, Czwallina-Allenstein 1996): Book I of this treatise begins with Postulate 1 describing the properties of a fluid at rest axiomatically. “Let it be supposed that the fluid is of such character that, its parts lying evenly (N.B.: at the same level) and being continuous (N.B.: coherent), that part which is thrust the less is driven along by that which is thrust the more and that each of its parts is thrust by the fluid which is above it in a perpendicular direction, unless the fluid is constrained by a vessel or anything else”. Although ARCHIMEDES does not use the word “pressure” and the Greeks did not know that concept in antiquity, he does infer that parts under more pressure would drive parts under less pressure so that a fluid cannot be at rest unless the pressure is uniform at a given depth, while the weight of a vertical column of fluid rests on the parts below it. From these very simple axiomatic premises, which do not permit evaluating the local pressure anywhere in the fluid, ARCHIMEDES is able to derive the principles of hydrostatic equilibrium and stability of floating bodies. This is achieved by considering the equilibrium of the resultant buoyancy and gravity forces and of their moments. ARCHIMEDES’ famous Principle of Hydrostatics is stated in Book I, Proposition 5: “Any solid lighter than a fluid will, if placed in the fluid, be so far immersed that the weight of the solid will be equal to the weight of the fluid displaced” (Archimedes 2002). The proof of this law, usually pronounced today as Δ = γ V, is brilliantly brief and conclusive. In all brevity it rests on the argument that in equilibrium the solid is at rest in a fluid at rest, thus if the body is removed from the fluid and the cavity left by its underwater volume is filled with fluid matter, then the fluid can only remain at rest if the replacing fluid volume weighs as much as the solid, else the fluid would not remain in equilibrium and hence at rest.
  • H. Nowacki and L.D. Ferreiro146 In Book II, mainly Proposition 2, ARCHIMEDES deals with the stability of hydrostatic equilibrium by treating the special case of a solid of simple shape, viz., a segment of a paraboloid of revolution of homogeneous material whose specific gravity is less than that of the fluid on whose top it floats. In equilibrium it floats in an upright condition. The stability is tested by inclining the solid by a finite angle to the vertical, but so that the base of the segment is not immersed. The equilibrium is defined as stable if the solid in the inclined position has a restoring moment tending to restore it to the upright condition. For the homogeneous solid this stability criterion is readily evaluated geometrically by examining the lever arm between the buoyancy and the gravity force resultants (Fig. 1). The buoyancy force acts through the centroid of the underwater volume B, which ARCHIMEDES finds for the inclined paraboloid from theorems proven earlier. The gravity force or weight acts through the Center of Gravity R of the homogeneous solid. Our conventional righting arm, the projection of BR on the horizontal, is positive. Instead of using this stability measure, ARCHIMEDES takes a shortcut for this homogeneous solid by splitting off and removing the weight of the submerged part of the solid Δ1 and the corresponding equal share of the buoyancy force, which have no moment about B because they both act through B. Thus only the weight of the abovewater section of the solid Δ2, acting through C, and the equal and opposite buoyancy force increment, acting through B, are taken into account. The centroid C is found from B and R by applying the centroid shift theorem when removing the underwater part from the system. This yields a positive “incremental righting arm” for the force couple of Δ2, acting through B and C respectively. The restoring moment is thus positive and the solid will return to the upright position. This application of the hydrostatic stability criterion by ARCHIMEDES is limited to the special case of a homogeneous solid of simple parabolical shape. It demonstrates the physical principles of the hydrostatic stability problem for a finite angle of inclination. It does not extend to floating bodies of arbitrary shape and of non-homogeneous weight distribution, hence actual ships. Still the foundations were laid to enable others much later to treat the generalized case of the ship on the same fundamental grounds. A more detailed appraisal of ARCHIMEDES’ contributions to the hydrostatic stability of floating systems and a thorough historical account is given by NOWACKI (Nowacki 2002).
  • Historical Roots of the Theory of Hydrostatic Stability of Ships 147 Fig. 1 Restoring moments, righting arms for inclined homogeneous paraboloid, based on ARCHIMEDES’ “On Floating Bodies”, Book II (Archimedes 2002) (figure from (Nowacki 2002)). 2.2 Stevin and Pascal In the beginning of the modern era of science the manuscripts of ARCHIMEDES had been rediscovered by several humanists during the Renaissance and were made accessible in print after 1500 by several editions in Greek, Latin and later by translations into modern languages (Archimedes 2002, Czwallina-Allenstein 1996, Dijksterhuis 1987). Simon STEVIN (born 1548 in Bruges, died 1620 in Leyden), the celebrated Flemish/Dutch mechanicist and hydraulic engineer, who knew and admired ARCHIMEDES’ works, was probably the first modern scientist who resumed and resurrected the study of hydrostatics and applied it to hydraulics, but also to ships. In his important work “De Beghinselen des Waterwichts” (Stevin 1955), first published in Dutch in 1586 and translated into Latin by Willibrord Snellius in 1605, he deals with the principles of hydrostatics and hydraulics. He adopts the idea of “specific gravity” of a fluid from ARCHIMEDES and introduces the concept of “a hydrostatic pressure distribution”, as we would call it today, proportional to the weight of a water prism reaching down to the depth in question. This enables him to calculate water loads on walls in a fluid, an important foundation in hydraulic engineering. He also rederives ARCHIMEDES’ Principle of Hydrostatics. However when he proceeds to examine the hydrostatic stability of a ship in his later note “Van de Vlietende Topswaerheyt” (Stevin 1586), he correctly reconfirms that for equilibrium buoyancy and weight force resultants must act in the same
  • H. Nowacki and L.D. Ferreiro148 vertical line through the centers of buoyancy (CB) and gravity (CG). But he erroneously concludes that for stability the CB must always lie above the CG. This error occurred because – unlike ARCHIMEDES - he neglected the centroid shifts resulting from the volume displacement from the emerging to the immersing side in heel. Despite this flaw in an application STEVIN deserves high recognition for founding modern hydrostatics on the concept of hydrostatic pressure. Blaise PASCAL (born 1623 in Clermont, died 1662 in Paris) is also counted among the founders of modern hydrostatics. He was familiar with ARCHIMEDES’ and probably with STEVIN’s work. In his “Traité de l’ équilibre des liqueurs”, Paris (1663), he arrives at similar conclusions and justifications regarding the fundamentals of hydrostatics in a fluid as STEVIN did, though he did not deal with ships. The assertion and experimental verification of hydrostatic laws also being applicable to air belong to his original contributions to this subject. 2.3 Huygens Christiaan HUYGENS (born 1629 in The Hague, died 1695 in The Hague), the eminent Dutch physicist, at the youthful age of 21 made an excursion into hydrostatic stability, which is not well known. In 1650 he wrote a three volume treatise “De iis quae liquido supernatant” (Huygens 1967a), in which he applies the methodology of ARCHIMEDES to the stability of floating homogeneous solids of simple shape, reconfirming ARCHIMEDES’ results and extending the applications to floating cones, parallelepipeds, cylinders etc., at the same time studying the stability of these solids through a full circle of rotation. He never published this work in his lifetime because he felt it was incomplete or “of small usefulness if any” or in any case not sufficiently original in comparison to ARCHIMEDES. He wanted the manuscript to be burnt, but it was found in his legacy and at last published in 1908. HUYGENS must be admired for his deep insights into ARCHIMEDES’ fundamentals of hydrostatics and for his own creative extensions. He recognized e.g. that for homogeneous, prismatic solids their specific weight and their aspect ratio are the essential parameters of hydrostatic stability. In his derivations he used a formulation based on virtual work as a principle of equilibrium. These examples of famous physicists between 1500 and 1700 working in hydrostatics and on hydrostatic stability of floating solids and ships demonstrate that the foundations inherited from ARCHIMEDES were understood by certain specialists, but had not yet been extended or applied to the design and stability evaluation of actual ships. Although the physical principles of stability were understood, the practical evaluation of volumes, volume centroids, weights and centers of gravity for floating bodies of arbitrary shape and non-homogeneous weight distribution de facto still posed substantial difficulties. These were not overcome before the advent and application of calculus during the 18th century.
  • Historical Roots of the Theory of Hydrostatic Stability of Ships 149 2.4 Hoste The French mathematics professor Paul HOSTE, s.j., (1652-1700) was the first to attempt to quantify the problem of ship stability. He did so in his 1697 treatise of naval architecture, Théorie de la construction des vaisseaux, which was appended to his book on naval tactics, written at the behest of Admiral TOURVILLE (Hoste 1697). However, he did not apply calculus to the problem, which was not yet widely known. HOSTE assumes, without citing STEVIN, that the CG and CB were in a vertical line; but he also allows without proof that the CG could be above the CB. However, to explain how this could be possible without the ship tipping over, he also assumes that the buoyancy force was equally divided between the two halves of the ship, forming the base of a triangle. He continues this error by stating: “If the center of gravity of the ship is known, the force with which it has to carry sail is easily known, which is no other thing than the product of the weight of the ship by the distance between these centers (of weight and displacement)” (Hoste 1697, p.54). In modern terms, the righting moment is equal to Δ x ( KG – KB ). In other words, the higher the center of gravity, the more stable the ship. Although he does not provide a theoretical means for determining this “power to carry sail”, HOSTE does describe a procedure which could empirically demonstrate this: The inclining experiment. HOSTE asserts that, by measuring the angle of inclination due to suspending a weight from a boom at a certain height, that the “force to carry sail” can be determined (Hoste 1697, p. 56). Although HOSTE’s argument contains several fundamental errors, it was the first attempt to express the stability of a ship in mathematical terms, and his book remained the only published inquiry into stability for almost half a century. 2.5 La Croix César Marie de LA CROIX (1690-1747) was the head of administration and finance for the Rochefort dockyard, and maintained the records for the galley fleet. He was not a scientist or engineer. Yet he developed some fundamental concepts on ship motions and hydrostatic stability, derived for a floating parallelepiped (La Croix 1736), so certainly before BOUGUER’s and EULER’s work appeared. LA CROIX was interested in the motions, but also in the hydrostatic restoring moments for this parallelepiped when heeled. He correctly understood the role of weight and buoyancy forces acting in the same vertical line in equilibrium. He also followed ARCHIMEDES in his stability criterion by examining the heeled body and requiring positive righting arms, accounting for the wedge volume shifts. But he falsely determined the influence of the wedge shift moments and
  • H. Nowacki and L.D. Ferreiro150 hence the righting arms. Besides, since he lacked integration methods based on calculus, he was not able to generalize his results for arbitrary section and waterplane shapes, hence to make them applicable to ships. Yet although LA CROIX’s work was flawed, it may have triggered EULER’s renewed interest in ship hydrostatics. In 1735 EULER was asked by the Russian Imperial Academy of Sciences to review the treatise (La Croix 1736), which LA CROIX had submitted there prior to publication. EULER quickly responded and appraised the merits of LA CROIX’s problem formulation, but also pointed out the weaknesses of the solution. He then put on record his own correct solution for the initial restoring moment, hence stability criterion, for the parallelepiped (or any other body of constant cross sectional shape), which he claimed to have found earlier (Euler 1972). This dates EULER’s earliest written mention of the criterion for this special case to 1735. In 1736 in reply to LA CROIX ’s rebuttal EULER also provided the complete explicit derivation of this result. 2.6 Sailing Vessel Propulsion and Maneuvering Questions of ship propulsion, ship motions and maneuvering, but also of ship stability have always aroused an acute interest, not only among seafarers and shipbuilders, but also in the scientific community. The propulsion of sailing vessels by the forces of wind, e.g., was investigated by ARISTOTLE in antiquity and later is associated with such celebrated names of scientists as Francis BACON, Thomas HOBBES and Robert HOOKE, among others. Toward the end of the 17th century the subject of the propulsion of sailing vessels had gained new scientific interest, also under the influence of the navies in applying scientific principles to ship design and operation. Treatises and monographs on the theory of maneuvering and on the equilibrium of aerodynamic propelling forces and hydromechanic response had appeared, notably by Ignace- Gaston PARDIES (1673), Bernard RENAU d’ELISSAGARAY (1689), Christiaan HUYGENS (1693), Jakob BERNOULLI (1695) and Johann (I) BERNOULLI (1714). This had led to lively controversies, but also to an increasing depth of understanding of the mechanics of the sailing vessel. In fact through the various treatises the principles for applying hydrodynamic underwater and aerodynamic sail forces to the vessel in order to establish its equilibrium position had become well understood. At the same time the importance of allowing for trim and heel (as well as yaw) under these forces and moments had been recognized and dealt with, although the quantitative assessment of the hydrostatic restoring effects remained largely an open issue. Both BOUGUER and EULER would begin their investigations into the nature of hydrostatic stability with the study of propulsion and maneuvering of sailing ships.
  • Historical Roots of the Theory of Hydrostatic Stability of Ships 151 2.7 Displacement Estimates Before Calculus The infinitesimal calculus was important to the development of stability for several reasons, including the practical evaluation of underwater volume and volume centroids. Yet a number of methods to calculate underwater volumes (and therefore displacements) were developed in the 1500s and 1600s, well before any theoretical framework for stability was available to make use of it. Why would shipbuilders go through the trouble of making such calculations? It appears that there were two separate reasons for this: The development of the gunport, and the measurement of cargo tonnage. The gunport was introduced in the early 1500s, and came into wide use by the middle of the century. This greatly increased the firepower of naval ships, but brought two problems; ships got much heavier, and with large holes in the side of the ship, the available freeboard got dramatically smaller. Thus, the margin for error in estimating the waterline was considerably reduced. To handle this problem, shipbuilders like Anthony DEANE (1638-1721) developed methods to calculate how much armament, ballast and stores should be loaded on a ship to bring it to the correct waterline below the gunports. DEANE and others made use of ship’s plans, which were just beginning to be employed by shipbuilders as a construction guide. In his manuscript “Doctrine of Naval Architecture” (Deane 1981), never published but widely circulated, DEANE demonstrates two methods to calculate the area underneath waterlines at each “bend” or frame of the hull; using either (1) an approximation for the area of a quarter-circle or (2) by dividing the area into rectangles and triangles. DEANE then sums the areas for each frame, multiplies by the frame spacing and multiplies the volume by the specific weight of water to obtain the displacement. He does this for several different waterlines, including the desired waterline below the gunports. When a ship is launched, he can immediately determine the light displacement, and then calculate how much weight should be added to arrive at the design waterline. A second reason for introducing displacement calculations was to more accurately measure the cargo capacity of a ship. For example, from 1646 to 1669 the Dutch and Danish governments carried on a series of negotiations on cargo measurement. In 1652 a Dutch mathematician Johannes HUDDE identified the basic issue of measuring cargo deadweight by determining the difference between the weight of the ship empty and fully-laden, using a difference-in-drafts method. He suggested to the Dutch authorities that they measure the waterplane areas at each draft of an actual ship in the water (not from plans), by taking measurements to the hull from a line extended at the side of the ship parallel to the centerline. The space between hull and an overall rectangle formed by the length and beam was then divided into trapezoids and triangles, the areas calculated and summed, multiplied by the difference in drafts and multiplied by seawater density, to obtain cargo tonnage. Although the suggestion was never used, HUDDE’s cousin
  • H. Nowacki and L.D. Ferreiro152 Nicolas WITSEN reported it in his 1671 shipbuilding manuscript “Aeloude en hedendaegsche scheepsbouw en bestier” (Witsen 1671; Cerulus 2001). 3 Development of Stability Theory 3.1 Bouguer 3.1.1 Biographical Sketch Pierre BOUGUER (Fig. 2) was born on 10 February 1698 in the French coastal town of Le Croisic, near Saint Nazaire at the mouth of the Loire. Educated in the Jesuit school in Vannes, he quickly showed himself a child prodigy. When his father Jean BOUGUER, a royal hydrographer and mathematician, died when Pierre was only 15, he applied for his father’s position. After initial hesitation by the authorities, he passed the rigorous exam and was given the post of Royal Professor of Hydrography. BOUGUER won several Royal Academy of Sciences prizes for masting, navigation and astronomy before he was 30. He moved to the port of Le Havre in 1731, about the same time he became a member of the Academy. His work caught the attention of the French Minister of the Navy MAUREPAS (1701-1781), who, like COLBERT before him, was convinced of the strategic benefit of ship theory as a way of compensating by quality against the quantitative superiority of the British navy. MAUREPAS supported BOUGUER’s research and sponsored his publications. In 1734, BOUGUER became involved in a controversy over the Earth’s shape; those who believed in DESCARTES’ vortex theories of physics held that the Earth was elongated at the poles, while those who accepted NEWTON’s theories of gravitational attraction argued that the Earth was wider at the equator due to centrifugal force. MAUREPAS also held the view that a full understanding of the Earth’s shape was essential to navigation, so he sent BOUGUER, along with several other members of the Academy of Sciences (accompanied by two Spanish naval officers) on a Geodesic Mission to Peru to determine the length of a meridian arc for a degree of latitude at the equator. One of his companions on the Mission was the young Spanish lieutenant Jorge JUAN Y SANTACILIA, who would later become a prominent naval constructor and author of a highly recognized treatise on naval architecture. BOUGUER spent ten years away (1735-1744), during which time he not only surveyed and calculated a meridian arc length of three degrees of latitude, he also performed various experiments on gravity and astronomy. It was during this Mission, in the peaks and valleys of the Andes far from the ocean, that he wrote much of his monumental work “Traité du navire”, the first comprehensive synthesis of naval architecture.
  • Historical Roots of the Theory of Hydrostatic Stability of Ships 153 Fig. 2 Portrait of Pierre BOUGUER. “Traité du navire” was published in 1746, shortly after BOUGUER’s return. He remained unmarried, lived in Paris and devoted himself to revising his geodesic work, and carrying out further studies of naval architecture, astronomy, optics, photometry and navigation. BOUGUER died in Paris on 15 August 1758, aged 60. Further details on BOUGUER’s biography and scientific work are presented by FERREIRO (Ferreiro 2007) and DHOMBRES (Dhombres 1999). 3.1.2 Early Work on Ship Theory In 1721 BOUGUER was asked by the Academy of Sciences in Paris to compare the accuracy of two methods of calculating cargo tonnage being proposed to the Council of the Navy for port fees. Although the mathematician Pierre VARIGNON proposed estimating the volume of a ship as a semi-ellipsoid, BOUGUER found that the best results came from a proposal of Jean-Hyacinthe HOCQUART of
  • H. Nowacki and L.D. Ferreiro154 Toulon, who used a difference-of-waterplanes method that employed equal-width trapezoids to estimate waterplane areas. This was similar to HUDDE’s approach but allowed a direct calculation of areas from ship’s plans. BOUGUER would later adopt and refine this “method of trapezoids” for his own stability work (Mairan 1724). In 1727 the Academy offered a Prize for the best treatise on masting, which BOUGUER’s entry won. He postulated that a “point vélique”, the intersection of the sail force and the resistance of water against the bow, should be directly above the center of gravity to minimize trimming by the bow. In his treatise he relied on HOSTE’s theories to explain stability, although where HOSTE implied that the advantage of doubling is through an increase in the center of gravity, BOUGUER invoked ARCHIMEDES to point out that the buoyancy of the added portion of the ship moves the center of buoyancy laterally when heeling, thus increasing the righting moment. Still, this treatise did not yet show any insights into the evaluation of trim or heel angles or restoring moments that he would develop five years later (Bouguer 1727). 3.1.3 The Metacenter BOUGUER probably began formulating his theory of stability around 1732, after he had moved to Le Havre, for he tested it using the little 18-gun frigate Gazelle, laid down in that dockyard in May 1732 and delivered in January 1734 (Demerliac 1995). However, no letters or manuscripts from that period survive to confirm this, and his derivation of the metacenter would not appear in print until 14 years later in “Traité du navire”. The following steps illustrate BOUGUER’s derivation of the metacenter in his “Traité du navire” (Bouguer 1746, pp. 199-324). STEP 1: Premises and axioms BOUGUER implicitly defines the hydrostatic properties of fluids, based on the principle of hydrostatics that weight and buoyancy are equal, opposite and act in the same vertical line. He does so without proof, without the use of equations and without mentioning ARCHIMEDES by name, although he shows general familiarity with his work. He also implies through geometrical arguments that the pressure of the fluid follows a hydrostatic distribution with depth and is everywhere normal to the surface of the submerged body.
  • Historical Roots of the Theory of Hydrostatic Stability of Ships 155 Fig. 3 BOUGUER’s diagram of the metacenter (Bouguer 1746, fig. 54). He denotes: p = submerged volume Γ = upright CB γ = inclined CB AB = upright waterline ab = inclined waterline I = CG of an unstable ship G = CG of a stable ship g = metacenter 1 = centroid of immersed wedge 2 = centroid of emerged wedge 3 = centroid of body without wedges STEP 2: Magnitude of buoyancy force BOUGUER resolves the submerged surface into very small elements (though without using any calculus notations at this point) and equates the vertical components of the hydrostatic pressure forces to the weight of the water column resting on top of the element in the interior of the submerged volume conceptually. Hence, the total pressure resultant is equal to the total weight of water filling the submerged volume. He thus reconfirms the law of ARCHIMEDES by pressure integration. STEP 3: Measurement of volume and centroids BOUGUER first suggests two methods for calculating the volume of the ship as a regular solid. The first is to model the ship as an ellipsoid, as originally proposed by VARIGNON, and the second is to divide it into prisms. The set of quadrilateral prisms then forms a polyhedral approximant of the hull surface for volume summation. BOUGUER sees clearly that, by analogy, the same principle of
  • H. Nowacki and L.D. Ferreiro156 polygonal approximation also holds for evaluating the area of planar curves. He thus quickly homes in on a quadrature rule for curves based on equally wide trapezoids, his favorite rule, also known as trapezoidal rule. He uses it to measure areas of waterline “slices” and then combining those two-dimensional slices to obtain a three-dimensional volume. Although he borrowed the idea from HOCQUART’s 1717 proposal, BOUGUER refined the method, first by dividing each waterplane into many sections (HOCQUART only proposed three sections), and second by taking the areas of several waterlines to develop the entire volume of the hull (HOCQUART took only one “slice”). BOUGUER then arrives at the conclusion that his quadrature rule is suitable for evaluating the integral of any continuous function of one variable or, if applied recursively, for continuous functions of any number of independent variables. Interestingly he thus interprets the analytical formulation of stability criteria, which he introduces later, as being equivalent to the discretized quadrature rules presented here earlier. The ancestry of his concepts from both practical shipbuilding traditions and modern calculus is still leaving some visible traces. BOUGUER then digresses for many pages into using the trapezoidal rule to calculate incremental waterlines for estimating a ship’s payload capacity, and gives various rules for tunnage admeasurement. For finding the centroid of the underwater volume, the center of buoyancy CB, BOUGUER then begins by explaining, in very simplistic terms, how to use the sum-of-moments method to determine the centroid of an object. He then derives the area centroid of a planar figure (2D case), then the volume centroid of a solid (3D case), which for a ship he calls the “center of gravity of the hull”. He then discusses by example how to evaluate these expressions numerically by the method of trapezoids. STEP 4: Stability criterion The center of buoyancy having been determined, BOUGUER next explains why he chooses the metacenter as the initial stability criterion, using the geometrical argument shown in Fig. 3. The ship’s center of gravity g is always in the same vertical line as the center of buoyancy Γ, but this geometry is not constant due to the ship’s movement. If the ship has a very high center of gravity I, and moves even a little from the upright position (waterline A-B) to another position (waterline a-b), it is no longer stati- cally stable; the center of buoyancy moves from Γ to γ, i.e., away from the vertical of the center of gravity, and the vertical force of buoyancy shifts from Γ-Z to γ-z. The ship’s weight, centered at I and on the opposite side of the inclination from the new center of buoyancy, tends to push the ship even further over, rendering it unstable. However, if the center of gravity of the ship is at G, below the intersection g of the upright and inclined vertical forces of buoyancy, then the center of gravity is on the same side as the new center of buoyancy and the resulting force always tends to restore the ship to the horizontal. BOUGUER states in his definition of the metacenter (Bouguer 1746, pp. 256- 257):
  • Historical Roots of the Theory of Hydrostatic Stability of Ships 157 “Thus one sees how important it is to know the point of intersection g, which at the same time it serves to give a limit to the height which one can give the center of gravity G, it [also] determines the case where the ship maintains its horizontal situation from that where it overturns even in the harbor without being able to sustain itself a single instant. The point g, which one can justly title the metacenter [BOUGUER’s italics] is the term that the height of the center of gravity cannot pass, nor even attain; for if the center of gravity G is at g, the ship will not assume a horizontal position rather than the inclined one; the two positions are then equally indifferent to it: and it will consequently be incapable of righting itself, whenever some outside cause makes it heel over.” BOUGUER does not use the terms “stable” or “unstable”, but rather states that G is either lower or higher than g. Step 5: Evaluation of the criterion The determination of this point of intersection, BOUGUER says, reduces to the question of the distance between the centers of buoyancy Γ and γ of the submerged body upright and just slightly inclined. BOUGUER employed the all- important shifting of equal immersed and emerged wedges to determine this distance, by drawing a triangle between points 1, 2 and 3. Since the centers of buoyancy Γ and γ lie on the legs of that triangle, the distance Γ – γ must be proportional to the distance between points 1 and 2, and the distance between Γ and point 3 must be proportional to the ratio of the volume of the underwater hull and the wedges. This geometrical explanation sets the stage for the second half of the explanation, the mathematical analysis of the mechanics of stability. Step 6: Results In the next section, BOUGUER uses calculus to determine the three unknowns: The distance from point 1 to point 2; the volume of the wedges; and the volume of the hull. BOUGUER imagines that Fig. 3 represents the largest section of the ship, which actually extends through the plane of the page in the longitudinal direction x, with the immersed and emerged wedges actually an infinite sequence of triangles of width y (the largest being b = F-B, or the half-breadth of the hull at the waterline) and height e ( = H-B) going through the length of the hull at a distance dx from each other; integrating, the volume of the wedge is Vwedge = b e 2 ∫ y2 dx (1) The second unknown is the distance between the centers of the wedges, points 1 to 2; but since the center of a triangle is two-thirds the height from apex to base, it is straightforward to obtain this distance as
  • H. Nowacki and L.D. Ferreiro158 Distance 1- 2 =   dx2y3 dx3y4 (2) The third unknown, the volume of the hull p, is derived using the trapezoidal rule. Putting the three together, the distance is: Γγ = bp3 dx3ye2  (3) Finally, observing that the triangle Γgγ (i.e., between the centers of buoyancy and the intersection of their vertical lines of force) is similar to the triangles formed by the immersed and emerged wedges, the height of the metacenter g above the center of buoyancy Γ is found via Euclid to be: Γg = p3 dxy2 3  (4) This is the now-famous equation for the height of the metacenter. 3.1.4 Implications of the Metacenter BOUGUER develops in thorough detail both the theoretical and practical aspects of the metacenter. He derives the metacenter for various solids (ellipsoid, parallelepiped, prismatic body) and presents the procedures for its practical, numerical calculation for ships. These explanations were detailed enough for practical applications and became the foundation for later textbooks. But BOUGUER also charged ahead beyond the initial metacenter for infinitesimal angles of heel when he introduced the concept of the “métacentrique”, i.e., the metacentric curve for finite angles of heel. First, he clearly recognized that the same physical principles and stability criteria apply to an inclined position of the ship as they do for the upright case. Second, he recognized that the metacentric curve for finite angles is in fact the locus of the centers of curvature of the curve of the centers of buoyancy. This was a brilliant, original insight. The locus of the centers of curvature of a curve was known since Christiaan HUYGENS’s work on the pendulum clock (Huygens 1967b) under the name of “développée” (or evolute). Third, BOUGUER also knew how to construct the evolute of a given curve. For a wall-sided ship (or parallelepiped) the metacentric curve is a cusp shaped curve composed of two hyperbolas lying above the metacenter, as BOUGUER showed. His demonstration examples for the “métacentrique” do document that he understood
  • Historical Roots of the Theory of Hydrostatic Stability of Ships 159 how to approach stability for finite angles of heel, though he never used a “righting arm” criterion. On practical aspects of initial stability he recommends that the widest point of the ship be no lower than where the maximum heel would be before it begins to tumble-home, or even that ship sides be straight or flared throughout. In other words, BOUGUER was advising to avoid tumble-home altogether, although in somewhat oblique terms. BOUGUER next provides a numerically-worked example of the application of the metacenter to a real ship, underlining the importance of Gg, the distance between the center of gravity and the metacenter. Using the Gazelle as his model, BOUGUER explains how to account for the weight of each part of the ship, including the frames, planking, nails, etc., how to measure the center of gravity of each piece using the keel as the reference point, and how to sum the moments to obtain the overall center of gravity G. He then calculates displacement and center of buoyancy of the ship using the trapezoidal rule, which enables him to find the metacenter g. His calculations confirmed that, once properly ballasted, the Gazelle would be stable. BOUGUER continues for almost 50 pages to outline the practical implications of the metacenter on hull design and outfitting. In many cases the implications are re-statements of what constructors already knew; but BOUGUER provided for the first time a rigorous analysis of why they were true. Some of the main points he brings out are:  The greatest advantage to stability lies in increasing the beam. BOUGUER claims that the “stability” varies as the cube of the beam, by which he must have been referring to the transverse moment of inertia I T, while the metacentric height requires closer scrutiny. But he wanted to emphasize the rapid increase of stability with beam. This also explains for the first time why doubling a ship improves stability, although BOUGUER does not explicitly state so.  Stability is improved by diminishing the weight of the topsides; though this was well-understood, BOUGUER details how to accurately assess the effects. BOUGUER also correctly describes for the first time how to evaluate the inclining experiment, for whose basic idea he credits HOSTE, in order to determine Gg, the distance between the center of gravity and the metacenter. In Fig. 4, he demonstrates how to use a known weight suspended from a mast to incline the ship and measure the angle of heel. Using the law of similar triangles, he shows that the ratio of the ship’s displacement and the suspended weight is proportional to the ratio of Gg to the angle of heel and distance the weight moves; on the assumption of small angles of heel this allows Gg (i.e., the reserve of stability) to be calculated directly without knowing the exact position of the metacenter. BOUGUER demonstrates how to use his metacenter by detailing the calculations of weights and centers of gravity for Gazelle while it was still being framed out. It is therefore curious that he did not verify this by performing an inclining experiment on Gazelle. It would appear that, while BOUGUER was completing his initial stability work in 1734, he was caught up in the events that led to the
  • H. Nowacki and L.D. Ferreiro160 Geodesic Mission, and would not return to the subject until he was in the mountains of Peru. Fig. 4 BOUGUER’s diagram of an inclining experiment, (Bouguer 1746, fig. 55). 3.2 Euler 3.2.1 Biographical Sketch Leonhard EULER (Fig. 5) was born in Basel, Switzerland on 15 April 1707. He was the son of Paulus EULER, parson of the Reformed Church, and his wife Margaretha née BRUCKER. He went to a Latin grammar school in Basel and, as his father recognized his talent early, took private lessons in mathematics. In 1720 he enrolled at the University of Basel as a student and later as a young scientist, received a Magister’s degree, then in theology. But he soon turned his main interest to mathematics and mechanics, which he studied under Johann (I) BERNOULLI, who was recognized as a leading mathematician of that era. In fact Johann (I) BERNOULLI, who was 40 years EULER’s senior, saw EULER’s maturing genius and invited him to join the Saturday afternoon “privatissimum” in mathematics, a private circle held in the BERNOULLI’s home, where he also met and made friends with younger members of the BERNOULLI family, notably Niklaus (II), Daniel (I) and Johann (II) BERNOULLI. In this period Johann (I) BERNOULLI laid the foundations from which EULER’s later mathematical fame developed. In 1727 EULER, who had been looking for an academic position, received an offer from the Russian Imperial Academy of Sciences in St. Petersburg, which Tsar Peter had initiated and which had been opened in 1725, where Niklaus (II) and Daniel (I) BERNOULLI held appointments as professors from this beginning. EULER accepted this prestigious offer, starting as an Adjunct (élève) for a modest in the first three years in philosophy where he
  • Historical Roots of the Theory of Hydrostatic Stability of Ships 161 salary, and arrived there – after a three month voyage by coach and by ship – in June 1727. EULER began his scientific career in St. Petersburg very successfully, advanced to a professorship in physics and full membership in the Academy in 1731, and spent his “First St. Petersburg Period” (1727-1741) very productively, working on a wide range of scientific subjects and publishing more than 50 treatises and books. In 1741, amidst political transitions and uncertainties in Russia, EULER received an attractive offer from King FREDERICK II of Prussia to come to Berlin and to work for the Royal Academy of Sciences, which was to be founded. EULER moved there in 1741, continued his illustrious scientific work on an increasing variety of subjects, and when the Academy at last opened in 1746 soon became recognized as one of its most eminent members and as the leader of the mathematical class. He remained immensely productive and published some of his most famous books and treatises during his period in Berlin from 1741 to 1766. Despite EULER’s scientific fame and the considerable merits he earned in the Berlin Academy during its formative first two decades he never developed a relationship with King FREDERICK that made him feel recognized, appreciated and understood. This was one contributing factor why after 25 years in Prussian service he decided to return to St. Petersburg. There he was welcomed and honored at all levels, and spent his “Second St. Petersburg Period (1766-1783)” in relative comfort, peace and recognition despite his weakening health and fading eyesight. His scientific creativity never ceased. A large share of his more than 800 treatises and books also stems from this second period in Russia. His work encompasses the full gamut of scientific topics in his era, not only in mathematics and mechanics, but also in other branches of physics, astronomy, ballistics, music theory, philosophy and theology. He died in St. Petersburg in 1783. More detailed accounts of EULER’s life and scientific vita are given by BURCKHARDT et al. (Burckhardt et al 1983), FELLMANN (Fellmann 1995) and CALINGER (Calinger 1996). 3.2.2 Early Work in Ship Theory In 1726 the Académie des Sciences in Paris invited contributions to a prize competition on the optimum placement, number and height of masts in a sail propelled vessel. EULER as a student and protégé of Johann (I) BERNOULLI was of course familiar with the earlier treatises and disputes that had arisen about the forces acting on a sail propelled maneuvering vessel between RENAU, HUY- GENS, Johann (I) BERNOULLI a.o. So he felt well enough prepared, also encouraged by Johann (I) BERNOULLI, to submit a treatise (“De Problemate Nautico…”) (Euler 1974) in 1727, which gained honorable mention, an “accessit” equivalent to a second prize. Pierre BOUGUER was awarded the first prize. In this treatise EULER, who still treads in the foot-steps of Johann (I) BERNOULLI, does not yet show deeper insights into ship hydrostatics although he does draw
  • H. Nowacki and L.D. Ferreiro162 attention to the requirement that for a ship before the wind the propelling sail force is limited by the acceptable forward trim angle. But he has no reliable physical basis for estimating that angle. Yet this early experience in his life at the age of 20 gave EULER a first thorough acquaintance with the mechanics of ships, familiarized him with nautical and ship construction terminology and created in him a lasting interest in ships as topics to be studied by methods of mathematical analysis and engineering mechanics. Fig. 5 Portrait of Leonhard EULER. In St. Petersburg EULER soon had ample opportunities to return to these subjects. It is not clear when he first occupied himself more deeply with ship hydrostatics. But evidently the appearance of LA CROIX’s treatise on the hydrostatic stability of a parallelepiped in 1735 found EULER well prepared, when the Academy asked for his review, to detect certain flaws in LA CROIX’s derivations and to present the correct result for this simple shape, which first established EULER’s initial restoring moment criterion on hydrostatic stability. In his second review in 1736, in response to LA CROIX’s rebuttal, EULER went beyond his first results, dealt with some other prismatic shapes (trapezoidal and triangular prisms) and
  • Historical Roots of the Theory of Hydrostatic Stability of Ships 163 alluded to his general approach to stability (Euler1972). This evidence is sufficient to date EULER’s earliest written mention of the hydrostatic stability criterion. There is also some indirect evidence of EULER’s earliest results on hydrostatic stability in his correspondence with the Danish naval constructor and naval attaché in London, Friderich WEGERSLØFF, which he maintained between 1735 and 1740. In a letter of 14 September 1736 WEGERSLØFF acknowledges receipt of a solution for the hydrostatic stability problem, on 22 May 1738 EULER replies and gives a derivation of his stability results and also mentions some experimental verification (Euler 1963). It is not clear from the context what sort of tests he had performed. By 1737 EULER’s interests in ship theory were well known at the Imperial Academy. Thus - probably not without his own prior knowledge or suggestion - he was commissioned by the Academy to write a book on this subject, resulting in his monumental two-volume “Scientia Navalis”. This work encompasses a presentation of the complete scope of ship theory according to the state of the art at that time and includes many new findings and derivations by EULER. The chapters in the two parts deal with ship hydrostatics, ship resistance and propulsion, maneuvering and ship motions. Many results have been of lasting value and have served as foundations for the growing scientific body of ship hydromechanics. According to EULER’s own report in the Preface of “Scientia Navalis”, he had worked on this manuscript from 1737 to 1740 (Euler 1967, pp. 15/16). When he left St. Petersburg in 1741 he had completed the first part, which contains all four chapters on hydrostatic equilibrium and stability, and half of the second part. The remaining sections of Part 2 were finished by 1741 in Berlin. Unfortunately, he had much difficulty finding a publisher for this voluminous opus, which thus was not published until 1749 by the Academy in St. Petersburg. Comprehensive appraisals of EULER’s overall contributions to ship theory and related matters are given by HABICHT (Habicht 1974), MIKHAILOV (Mikhailov 1983) and TRUESDELL (Truesdell 1983). 3.2.3 The Initial Stability Criterion To understand the history of EULER’s involvement in ship theory it is important to read the Preface to his “Scientia Navalis”. Here he explains that his work will go beyond the established disciplines of hydrography or nautical science and will concentrate on a physical and analytical investigation of the mechanics of ships, at rest and in motion, for which fundamental theoretical works were still missing at that time. In hydrostatics EULER departs from the Principle of ARCHIMEDES, to whom in the preface of “Scientia Navalis” he gives credit and praise. But he adds that the hydrostatic stability of ships must be newly approached and quantified in order to be able to distinguish between stable and unstable equilibrium of ships at the design stage. The experience of naval architects (“architecti navales”) alone, long
  • H. Nowacki and L.D. Ferreiro164 established as it may be, will not be sufficient to prevent unexpected stability accidents. EULER nowledges the motivation he received from reviewing LA CROIX’s treatise in 1735-1736, which prompted him to investigate more profoundly the transverse and longitudinal stability of ships. EULER also gives very favorable credit to BOUGUER’s “Traité du Navire”, which had appeared 3 years before “Scientia Navalis”, but he takes much care also to avert any suspicion of plagiarism by calling on the Imperial Academy as witnesses that he had written “Scientia Navalis” essentially between 1737 and 1741 during which period there was no communication with BOUGUER who was in Peru. These statements were never disputed between the two authors, who corresponded in a respectful and amicable fashion on other subjects later. EULER’s derivation of his stability criterion in “Scientia Navalis” proceeded in the following steps (Euler 1967, pp. 3-166): STEP 1: Premises and axioms In the first chapter of Book I, EULER deals with the equilibrium of bodies floating in water at rest. In essence he rederives the Hydrostatic Principle of ARCHIMEDES from the modern viewpoint of infinitesimal calculus by integration of the hydrostatic pressure distribution prevailing in a fluid over the surface of the body. The properties of the fluid at rest and the use of calculus for this purpose both were new at the time when EULER wrote these passages. As for the pressure distribution he states axiomatically in the opening paragraph of his book: “Lemma: The pressure which the water exerts on the individual points of a submerged body is normal to the body surface; and the force which any surface element sustains is equal to the weight of a straight cylinder of water whose base is equal to the same surface element and whose height is equal to the depth of the element under the water surface”. These brief axioms, viz. the normality of pressure to a surface and the inferred equality of the pressure at a point in a given depth in all directions, were the first analytical formulation for the properties of the fluid and are regarded as the necessary and sufficient conditions for the foundation of hydrostatics (Truesdell 1954). STEP 2: Magnitude of buoyancy force From these premises EULER proceeds to define by integral calculus the buoyancy force as the pressure resultant and the center of buoyancy (EULER calls the CB: “centrum magnitudinis”), through which it acts, as the volume centroid of the submerged part. He reconfirms also that for equilibrium the buoyancy and weight forces must act in the same vertical line and must be equal in magnitude and opposite in direction. He then illustrates these principles by examples of simple shapes like parallelepipeds and prisms of triangular and trapezoidal cross section. For each of these solids he finds the possible equilibrium conditions over a full circle of rotation as a function of the specific weight of these homogeneous solids, not unlike HUYGENS in his unpublished treatise of 1650. ack
  • Historical Roots of the Theory of Hydrostatic Stability of Ships 165 In Chapter II, briefly digressing from hydrostatics, EULER discusses the resulting motions of a floating body if it is temporarily displaced from its “upright” equilibrium position. Here he explains the “lumping” of masses in their center of gravity (CG), introduces the definition of the principal axes of inertia for ships, underscores the significance of the CG as a reference point, e.g., for decomposing a resulting motion into the translation of the CG and the rotation about it. EULER will adhere to the CG as his system reference point, also when returning to ship motions later. STEP 3: Measurement of volumes and volume centroids In stark contrast to BOUGUER, EULER confines himself to analytical definitions of stability criteria, volumes, centroids, areas, moments of inertia etc. He does not address their numerical evaluation at all. He is taking it for granted that once the shape of the ship or body is defined by some function the integrations can be readily performed. In his examples he usually deals with simple shapes where the integrations can be performed in closed form. STEP 4: Stability criterion In Chapter III, the main chapter on hydrostatic stability, EULER defines his stability criterion right away (Proposition 19) by: “The stability which a body floating in water in an equilibrium position maintains, shall be assessed by the restoring moment if the body is inclined from equilibrium by an infinitesimally small angle”. EULER illustrates this principle by discussing cases of unstable, neutral and stable equilibrium of ships and adds that it is necessary to quantify stability in terms of the restoring moment because even a stable ship may be in danger by external heeling moments and may require righting moments of greater magnitude. (The issue is not only whether the ship is initially stable or not, but also how much “stability capacity” it has). STEP 5: Evaluation of the criterion EULER then enters into the determination of the restoring moments by first examining a planar cross section of arbitrary shape (or thin disk) floating upright (Fig. 6). The Figure AMFNB = “AFB” is inclined by a small angle dw so that the new floating condition has the waterline ab. Since the center of gravity G remains his reference point, also for later purposes, he draws the parallel lines MN and mn to the waterlines before and after inclination, also through G. The center of buoyancy of the cross section is designated by O. A normal VOo to the inclined waterline is drawn through O. The equality of the immersed and emerging wedges requires that ab intersect AB at the center point C so that AC = BC.
  • H. Nowacki and L.D. Ferreiro166 Fig. 6 EULER’s figure for centroid shift in inclined cross section, 2D case, (Euler 1967, fig. 39). Let the displacement per unit length and the equal buoyancy force of the cross section before inclination be denoted by M = γ (AFB). For the inclined position the restoring moment is composed of three contributions: 1. The effect of the original submerged volume forming a positive restoring couple of forces through G and V: M GV = M GO dw (5) 2. The effect of the submerged wedge CBb whose cross section area is: 2 dwBC2 = 8 dwAB2 (6) and whose restoring moment about G hence is: γ 2 dwAB2 (qo + GV) (7) where qo = 3 2 Cb = 3 1 AB 3. Likewise for the emerging wedge ACa the restoring moment is -γ 8 dwAB2 (po - GV) (8) where po = 3 2 Ca = 3 1 AB For all moments combined, replacing γ by M/ (AFB):
  • Historical Roots of the Theory of Hydrostatic Stability of Ships 167 M REST = M GO dw + AFB8 )qopo)(dwAB(M 2  = (9) The expression in square brackets has the dimension of a length and is the now well known result, corresponding to, in BOUGUER’s later terminology, the metacentric height of the cross section. This is how EULER’s restoring moment and BOUGUER’s metacentric height are connected. Note that the GO term reverses sign if G lies above the center of buoyancy O, as is common in cargo ships. EULER discusses this result at some length and for several simple shapes. In Proposition 29 EULER then arrives at the general three-dimensional case of a floating body of arbitrary, in particular asymmetrical shape, whose waterplane is drawn in Fig. 7. He denotes: M = displacement or weight of body V = submerged volume GO = distance between CG and CB, positive for G above B CD = reference axis of waterplane through centroid of waterplane area, parallel to axis through G CX = x = abscissa from origin C XY = y = ordinate in upper part of waterplane XZ = z = ordinate in lower part of waterplane p and q = area centroids of upper and lower parts of waterplane P and Q = equivalent pendulum lengths of waterplane area parts w.r.t. axis CD With pr =   ydx dxy 2 2 qs =   zdx2 dxz2 PR =   dxy dxy 3 2 2 3 QS =   dxz dxz 3 2 2 3 Mdw [GO + AFB12 AB3 ]
  • H. Nowacki and L.D. Ferreiro168 Fig. 7 EULER’s figure for the derivation of the stability criterion for a body of arbitrary shape, 3D case (Euler 1967, fig. 48). EULER derives, since the axis CD runs through the waterplane centroid: ∫ y2 dx = ∫ z2 dx, and hence PR + QS = 3 3 2 2 3 ( )y z dx y dx   (10) from which in analogy to the planar case the restoring moment, divided by the angle dw, for the ship becomes: M (GO + V3 dx)zy( 33   ) (11) In the special case of port/starboard symmetry (y = z), with IT = 3 2 ∫ y3 dx; and TI V = OM In our familiar notation (GO= GB , OM= BM ) the restoring moment simplifies into: M ( GB + BM ) = M GM (12)
  • Historical Roots of the Theory of Hydrostatic Stability of Ships 169 STEP 6: Results This summarizes the course of steps EULER took to derive the initial restoring moment. EULER never used the word metacenter. EULER’s result thus is the initial restoring moment, divided by (M dw), which he uses as his stability criterion. Again EULER illustrates this result by many examples for simple shapes of solids, even by an analytical formulation for a shiplike body with parabolic section shapes. But he does not present any numerical calculations for an actual ship although he discusses many practical implications of his results. 3.2.4 Implications of the Stability Criterion EULER interprets his results for practical applications to ships, mainly in Part II of “Scientia Navalis”. For EULER ships are “floating objects, carrying a cargo or payload, symmetrical starboard and port, propelled by rowing and/or sails”. Among the major conclusions he draws, we quote: Stability is judged by the restoring moment divided by the angle dw. Transverse and longitudinal stability are clearly distinguished and both addressed. EULER also derives an expression for combined heel and trim under oblique sail force moments, deriving the oblique restoring effects by rotation of the reference axis from the principal axes. To assess stability you need to know the ship’s displacement, the centroids CB and CG, the waterplane area and shape, which infers the restoring moment (without using the metacenter). To improve stability, lower the CG, raise the CB and/or widen the beam. The addition, removal and internal shifts of weight as well as the role of ballast for stability are discussed. The use of doubling (“soufflage”) with its pros and cons is mentioned. In Chapter IV of Part 1 EULER addresses practical problems of inclining ships under external moments or by internal weight shifts, though again only analytically, not numerically. He also addresses the issue of external wind loads and required stability, especially for small ships vs. big ships, cargo ships vs. war ships. EULER finally also makes the interesting “philosophical” remark that sailors knew about measures of stability all along. When the sailors say, “This ship can sustain a strong wind in its sails”, then EULER claims they mean the same thing as he expresses by his restoring moments. 3.3 Comparison of Approaches Based on the facts presented above it is now possible and in fact intriguing to compare the approaches taken by BOUGUER and EULER in their creative work on hydrostatic stability. This will bring out certain commonalities, but also underscore the differences. We have examined several aspects.
  • H. Nowacki and L.D. Ferreiro170 3.3.1 Chronology: BOUGUER worked on the subject of ship hydrostatic stability from about 1732 to at least 1740, and his treatise was published in 1746. EULER was engaged with this topic between about 1735 to at least 1740, “Scientia Navalis” appeared in 1749. During these formative periods of their new concepts there existed no contacts and no communications between them. Thus their work originated independently. But after their first scientific acquaintance through their participation in the 1727 Paris Academy award competition it is likely that they both underwent a common fermentation period on related subjects. Probably they were both looking for the missing pieces in the mosaic of sailing ship maneuvering, among which the hydrostatic response was foremost. It appears that their deeper familiarity with ARCHIMEDES originated during the period between 1727 and 1735. 3.3.2 Sources: BOUGUER writes that he was familiar with the work of PARDIES, HOSTE, RENAU, most likely also with relevant publications by HUYGENS and the BERNOULLIs. He shows close acquaintance with ARCHIMEDES’ principles, although he does not mention his name. On calculus he quotes the well-known treatise by NEWTON and Roger COTES, but very likely he also thoroughly read de l’HÔPITAL, whose notation in the LEIBNIZ style he preferably uses. His knowledge of calculus was mostly self-taught, based on occasional tutorials and recommendations from mathematics professor Charles René REYNEAU, who wrote the basic textbook on the subject, “Analyse démontrée” in 1708. EULER also knew ARCHIMEDES, HOSTE, RENAU, whom he mentions. He learnt calculus first-hand from the BERNOULLIs, who stood firmly in the LEIBNIZ tradition. EULER’s book on mechanics, “Mechanica” in 2 volumes, where he displays exquisite knowledge of the principles of equilibrium and motions of mechanical systems, had appeared in 1736. Although both authors are not very generous in giving references on their sources it is possible to a certain extent to trace their intellectual ancestry by looking at their scientific and technical terminologies, even if they wrote in different languages, French and Latin. We have analyzed their vocabularies in their main treatises on hydrostatic stability and identified commonalities and distinct differences. They shared much common ground by using such established words as equilibrium, stability, weight, buoyancy, specific weight, centers of gravity and buoyancy, inclination (derived from ARCHIMEDES) and pressure, hydrostatics (from STEVIN and PASCAL). BOUGUER is unique in inventing the metacenter and using the evolute (from HUYGENS), but never the word “restoring moment”, while EULER’s vocabulary was vice versa. This indicates where their approaches differed.
  • Historical Roots of the Theory of Hydrostatic Stability of Ships 171 3.3.3 Stability Criteria: Sections 3.1 and 3.2 have shown that BOUGUER and EULER derived equivalent stability criteria, but used different approaches. To sum up the comparison: Premises and axioms: BOUGUER’s work is founded on the premises of ARCHIMEDES, though augmented by a hydrostatic pressure law (STEVIN). EULER defines the axioms of hydrostatics a bit more strictly by postulating the direction independence of pressure and its normality to a surface. Magnitude of buoyancy force: Both agree and reconfirm the law of ARCHIMEDES by pressure integration and calculus. Measurement of volumes and centroids: BOUGUER takes the route of establishing numerical procedures, based on quadrature rules, for measuring volumes, centroids, areas etc. before giving analytical definitions of the stability criterion. In passing he skillfully adapts the “method of trapezoids” to shipbuilding applications. EULER skips this step entirely and drives directly at analytical formulations for his stability criterion. Stability criterion: BOUGUER’s invention and choice for a measure of stability is “metacentric height”, a geometric quantity representative of initial stability capacity. EULER chooses the “initial restoring moment”, a physical quantity, also measuring the ship’s initial stability capacity. Although the two ideas are closely connected, their meanings are conceptually different. BOUGUER never refers to “restoring moments”, EULER never uses “metacenter”. Neither author recurs to ARCHIMEDES’ idea of a “righting arm”. Evaluation of the criterion: BOUGUER derives the metacenter mainly from geometric arguments and, allowing for the shift of wedge volumes and hence CB, determines the metacenter as the point of intersection of the upright and inclined buoyancy directions. EULER uses physical arguments of moment equilibrium and shift of volumes and finds the restoring moment simply by a summation of moments. Results: BOUGUER’s result is metacentric height, a distance. EULER’s result is a restoring moment. The two results are equivalent in practice, but not equal in approach. 3.3.4 Audience, Style and Language: BOUGUER’s “Traité du navire”, the first modern synthesis of theoretical naval architecture, is written in French for a readership of scientists and constructors who are to be introduced to the use of theory in practical ship design and shipbuilding. BOUGUER benefited from both his experience as a hydrographer and his collaboration with ship constructors. His style is clear and logical, explaining many practical details, almost as in a textbook. He is concerned about how this methodology can be implemented. This style does not cause any sacrifices in rigor. His language is lucid, his own new ideas come across crisply.
  • H. Nowacki and L.D. Ferreiro172 EULER in his Latin text of “Scientia Navalis” writes as an applied mechanicist/ physicist and addresses mainly the scientific community of his era. Among those who were educated to read scientific treatises of this sort in Latin, which was still a lingua franca in the academic world, EULER’s brilliance of style, his inimitable logic and clarity were highly praised. These attributes also apply to his “Scientia Navalis”. His work on hydrostatic stability is still valid today. But at the same time he did not have any experience with practical shipbuilding applications. Thus he left many details for implementation by later successors, which delayed the spreading of his methods, certainly in comparison to BOUGUER. Yet in the final balance the scientific and engineering community must be grateful to both BOUGUER and EULER. 3.4 Synopsis Table 1 gives a synopsis of important milestones and ideas in the development of the theory of hydrostatic stability by indicating where certain elements of this theoretical knowledge first occurred. The importance of ARCHIMEDES’s physical insights and the relevance of BOUGUER’s and EULER’s parallel discoveries described by calculus become clearly visible in this condensed tabular summary of historical steps. Table 1. Historical development of stability concepts
  • Historical Roots of the Theory of Hydrostatic Stability of Ships 173 4 Further Work 4.1 Practical Applications Stability theory quickly found direct applications in the day-to-day practice of ship design. This occurred in two ways: first, by the increasing sophistication of calculations for weights, centers of gravity and the metacenter within the design process; and second, by the use of inclining experiments to validate stability. These applications were surprisingly widespread; stability theory was quickly incorporated in navies where there was already a strong institutional development of scientific naval architecture, notably in France, Denmark, Sweden and Spain. Those countries also created schools of naval architecture during the 18th century, where students were weaned on stability theory and naturally used it when they became constructors at the dockyards. This was not the case for the British or Dutch navies, which had provided little direct support for scientists working on ship theory, and which did not establish permanent schools of naval architecture until the 19th century. 4.1.1 Stability Calculations During Design The most famous of these schools of naval architecture was the Ecole du Génie Maritime, established in 1741 in France by the scientist Henri Louis DUHAMEL DU MONCEAU (1700-1782), whom MAUREPAS had appointed to Inspector- General of the Navy. DUHAMEL worked with BOUGUER to create the first comprehensive textbook for the students, based on “Traité du navire”. DUHAMEL’s genius was to take BOUGUER’s complex mathematics and render them into step-by-step instructions on how to calculate the metacenter. The textbook, “Elémens de l’architecture navale” (Duhamel 1752) became the standard reference for both students and constructors. DUHAMEL DU MONCEAU was also involved in developing the Ordinance of 1765 under Navy minister Etienne François, Duke of CHOISEUL, which created the Corps du Génie Maritime and formalized the data to be included on ship’s plans: “centers of gravity and resistance, and height of the metacenter” as well as accompanying calculations and a tabulation of hull materials. Later, the standardized plans introduced in 1786 by DUHAMEL’s successor as Inspector- General, Jean-Charles de BORDA, listed the specific immersion of the hull at full and light load in tonnes-per-cm equivalent, which allowed a rapid estimate of the effect of loading weights on the ship.
  • H. Nowacki and L.D. Ferreiro174 4.1.2 Inclining Experiments Another facet of rapid adoption of the metacenter was that the first recorded inclining took place in 1748, just two years after BOUGUER published his work. Francois- Guillaume CLAIRIN-DESLAURIERS, then a junior constructor at Brest, performed the experiment on the newly-built 74-gun ship Intrépide, apparently out of curiosity to test the new theory. He hung two 24-pound cannons (each weighing over 2 tonnes) from a buttress built on the side of the ship (Fig. 8), and although the buttress broke, he was able to take enough measurements to ascertain that the ship’s GM was 1.8m (Clairin-Deslauriers 1748). Fig. 8 CLAIRIN-DESLAURIERS’ drawing of the inclining of Intrépide 4.1.3 CHAPMAN and SIMPSON’s Rule Another important contribution to the promulgation of hydrostatic stability calculations came from the Swedish naval constructor and innovative ship designer Fredrik Henrik CHAPMAN. CHAPMAN (1721-1808), the son of the superintendent of the Göteborg dockyard, was endowed by this background with much practical shipbuilding knowledge. He visited Britain in 1750 to improve his knowledge in mathematics and other fundamentals. He took classes from Thomas SIMPSON (1710-1761), a gifted mathematician trained in NEWTON’s tradition and an instructor at the Royal Military Academy in Woolwich near London. SIMPSON reportedly taught CHAPMAN a quadrature rule for planar curves, which yields a numerical approximation of the integral of a continuous function using at least one equally spaced double interval, hence at least three offset points per double interval, or any desired number of further double intervals of the same spacing. This quadrature rule later became known as “SIMPSON’s Rule”, especially in shipbuilding, although the rule has several historical precursors and SIMPSON never claimed to have invented it. CHAPMAN, after further visits to France and Holland and having picked up much valuable background including (Clairin-Deslauriers 1748)
  • Historical Roots of the Theory of Hydrostatic Stability of Ships 175 the recent findings by BOUGUER on the metacenter, returned in 1757 to Sweden to continue his career as an important ship constructor. In this process he soon made it his habit to calculate ship displacements (1767), also as a function of draft, and assessing the metacenter, applying SIMPSON’s Rule for numerical integration. His engineering methods applied in ship design became well known internationally soon after the publication of his book Tractat om Skepps-Byggeriet (Chapman 1775), where he explained his calculations in much detail (Harris 1989). Thus CHAPMAN helped to promulgate the use of the metacenter and to make its calculation a routine matter among naval architects. At the same time SIMPSON’s Rule, which offers certain efficiency advantages over the trapezoidal rule for an equal number of intervals, was popularized and is still much favored today. 4.2 Extensions of Stability Theory 4.2.1 Hydrostatic Stability About Other Axes In their work on the masting of sailing ships in 1727 it must have occurred to both BOUGUER and EULER that in general the external wind moments would act in an oblique direction to the ship’s centerplane so that heeling and trimming would result simultaneously. EULER remembered this open question and in “Scientia Navalis”, Part 2, returned to this issue. He derived an expression for the restoring moment about an oblique axis between the axes of heel and trim. The moments of inertia about this axis could be constructed, when the moments of inertia IT and IL for transverse and longitudinal stability were known, by using an ellipse of inertia moment construction. Thus this method was able to predict at least the initial hydrostatic response (small angles) to a wind moment in an oblique plane in terms of simultaneous heel and trim angles. EULER mentioned on the side, too, that longitudinal stability was always much greater than transverse stability. It was some time later that the Spanish naval constructor and engineering scientist Jorge JUAN Y SANTACILIA (1713-1773), who knew BOUGUER and his work very well from the Peruvian expedition and also was in correspondence with several European Academies, concerned himself not only with ship stability, but also with ship oscillations, especially with pitching motions. In this context he extended BOUGUER’s concepts of the metacenter to longitudinal inclinations and first introduced the definition of the longitudinal metacentric height GML. He pointed out that it by far exceeds the transverse metacentric height. His book “Examen Maritimo”, which appeared in 1771, combined the theoretical insights of his days with much practical design experience and became a widely used reference and textbook (Juan y Santacilia 1771).
  • H. Nowacki and L.D. Ferreiro176 4.2.2 Large Angles of Heel A broader extension of stability theory occurred not on the European continent but in Britain. In two papers presented to the Royal Society of London, the British mathematician George ATWOOD examined the inclination of ships at large angles of heel. The first paper (Atwood 1796) based on a thorough knowledge of ARCHIMEDES, BOUGUER and EULER, reviews the fundamental physical principles of hydrostatic stability, applied to finite angles of heel. Here ATWOOD investigates the stability properties of homogeneous solids of simple shape (parallelepiped, cylinder, paraboloid) through 360 degrees of rotation as a function of body draft at rest (or specific weight). He finds numerous equilibrium positions (8 or 16), only some of them stable. He develops a shifted wedge volume method for finite heeling angles and reviews the numerical quadrature rules, settling for STIRLING’s 3 interval rule. This paper already drew full attention to the fact that stability must be judged over a range of finite, practical heel angles; initial stability alone is inadequate as a stability measure. In the second paper (Atwood and Vial du Clairbois 1798), which ATWOOD coauthored with the French constructor Honoré-Sébastien VIAL DU CLAIRBOIS, the investigation was extended to actual ships, and for the first time a numerical analysis of the “righting moments” of ships over a large range of heeling angles was performed. ATWOOD and VIAL DU CLAIRBOIS introduced the term GZ for the “righting arm”, which was again numerically evaluated by a wedge volume shift method. This successfully demonstrated the feasibility of numerical stability analysis for actual ships over a range of finite heeling angles. The necessity of performing such calculations rather than just relying on initial stability measures was again underscored. Yet in shipbuilding practice it still took several more decades before the initial difficulties in numerical integration could be overcome by more robust methods and instruments. Even half a century later such stability evaluation had not yet become a routine matter. It was not before the early 19th century that the knowledge on the metacenter progressed one further step by the work of French constructor and mathematician Charles DUPIN from metacentric curves to metacentric surfaces (Dupin 1813). DUPIN stated that the ship for finite angles of inclination in whatever direction possesses a surface of centers of buoyancy. This surface at every point has two principal curvatures, thus one can construct two sheets of metacentric surfaces, i.e., evolute surfaces of the buoyancy surface. The two metacenters for the upright condition, MT and ML, each is a special point on one of the surfaces. It is probably fair to say that at this level of perception the issue of hydrostatic stability had been fully resolved in terms of the relationships between ship geometry, physical responses to inclinations, and differential geometry of metacentric surfaces for finite angles of heel.
  • Historical Roots of the Theory of Hydrostatic Stability of Ships 177 5 Conclusions ARCHIMEDES laid the foundations for the stability of floating systems, introduced a stability measure similar to the righting arm and presented an approach for assessing the ability of a floating inclined solid to right itself. But his applications were limited to simple geometrical shapes. Fortunately his manuscript “On Floating Bodies” survived in a few copies and became more accessible again by a Latin translation in the 13th c. and also in print after 1500. Yet it took a few more centuries before modern hydrostatic stability was established and could be applied to actual ships, in particular also at the design stage. The interest in scientific solutions for ship stability gained new momentum for practical and scientific reasons by about 1700: The leading navies were getting more concerned about stability risks with increasing ship sizes, gunports that were close to the water and increasingly heavier armament. In the beginnings of the Age of Enlightenment, new expectations were raised with regard to the capabilities of science to predict physical phenomena and the performance of technical systems. Mathematical breakthroughs occurred in infinitesimal calculus, analytical and numerical methods. The modern age of engineering science made rapid progress in mechanics and hydromechanics, including the study of equilibrium and stability of mechanical systems. All of this contributed to an atmosphere in the scientific community in the early 18th c. that was open to new challenges, also in application to ships. Both BOUGUER and EULER were actively involved in these general developments and certainly exposed to this unique zeitgeist. BOUGUER responded to the recognized problem of hydrostatic stability more as an engineering scientist, EULER reacted rather more like an applied mechanicist and mathematician. Both were able to reformulate and solve this problem in their own unique and original ways. As our comparisons in this paper have reconfirmed, BOUGUER’s and EULER’s nearly simultaneous work was not only performed quite independently, which was never doubted, but was also distinctly different in approach and justification. Both investigated the ability of the ship to right itself after an infinitesimal heeling displacement. BOUGUER reasoned mainly geometrically and for the inclined ship derived a length measure, the metacentric height, as the decisive geometric measure of initial stability. EULER argued mainly on mechanical grounds and deduced the initial restoring moment as his measure of hydrostatic stability. Despite these contrasting styles of justification, as we appreciate today of course, both stability measures are in fact equivalent and can be converted into each other. We may still benefit from both lines of thought and should be grateful for the insights we owe to these two congenial pioneers.
  • H. Nowacki and L.D. Ferreiro178 We have explained why it took several more decades before these landmark discoveries in ship stability were fully accepted and widely applied in shipbuilding practice. But the foundations laid by BOUGUER and EULER have remained of lasting value as secure cornerstones in our knowledge for designing safe and stable ships. Acknowledgments The authors are pleased to acknowledge the support and encouragement they received during the preparation of this work from many colleagues, in particular from Jean DHOMBRES, Emil A. FELLMANN, Jobst LESSENICH and Gleb MIKHAILOV. The first author also expresses his gratitude to the Max Planck Institute for the History of Science Berlin, where he is a Visiting Scientist and where his work was much promoted by personal communications and the excellent library resources. References Archimedes (2002) The Works of Archimedes, edited and translated by T.L. Heath, repub by Dover Publ, Mineola, N.Y. Atwood G (1796) The Construction and Analysis of Geometrical Propositions, Determining the Positions Assumed by Homogeneal Bodies which Float Freely, and at Rest, on a Fluid’s Surface; Also Determining the Stab of Ships and of Other Float Bodies, Philos. Trans. of the R. Soc. of London, Vol. 86: 46-278. Atwood G, Vial du Clairbois HS (1798) A Disquisition on the Stability of Ships, Philos. Trans. of the R. Soc. of London, Vol. 88, vi-310 Bouguer P (1727) De la mâture des vaisseaux, pièce qui a remporté le prix de l’Acad. R. des Sci. de 1727, (On the masting of ships) Jombert, Paris. Bouguer P (1746) Traité du Navire, de sa Construction et de ses Mouvemens (Treatise of the Ship, its Construction and its Movements), Jombert, Paris Boyer CB (1949) The History of Calculus and its Conceptual Development, 1939, reprint Dover Publ, N.Y. Burckhardt JJ, Fellmann EA, Habicht W, eds (1983) Leonhard Euler 1707-1783, Mem. Monogr., issued by the Canton Basel-City. Birkhäuser Verlag, Basel. Calinger R, Leonhard Euler (1996) The First St. Petersburg Years (1727-1741), Hist. Math., vol. 23, 121-166, Academic Press. Cerulus FA (2001-Jan) Un mathématicien mesure les navires (A mathematician measures the ships), in Géomètre; édition des Histoires de la mesure, Publi-Topex editions, Paris, 34-37. Chapman FH af (1968) Tractat om Skeppsbyggeriet (Treatise on Shipbuilding), Johan Pfeffer, Stockholm (1775); translation and facsimile with commentary in Architectura Navalis Mercatoria (Merchant Nav Archit), Praeger Publ, N.Y. Clairin-Deslauriers G (1748-May 8) Letter to Duhamel du Monceau, Peabody-Essex Museum, collection Duhamel du Monceau, MH 25 box 2 folder 2 item 4.8; Bres. Czwallina-Allenstein A (ed.) (1996) Treatises of Archimedes, translated and annotated by the editor in 1922, republished in Ostwald’s Klassiker der exakten Wissenschaften, vol. 201, Verlag Harri Deutsch, Frankfurt am Main.
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  • H. Nowacki and L.D. Ferreiro180 Huygens C (1967b) Horologium Oscillatorium…, (Pendulum Clockwork…), Part III, F. Muguet, Paris, 1673. In Œuvres Complètes de Christiaan Huygens, vol. 18 (1934). Swets and Zeitlinger, Amsterdam. Juan y Santacilia J (1771) Examen marítimo, Theórico Práctico (Maritime examination, theoretical and practical), Manuel de Mena, Madrid. La Croix CM de (1736) Eclaircissemens sur l’Extrait du méchanisme des mouvemens des corps flotans (Clarifications on the extract of the mech of the movements of floating bodies), Robustel, Paris. Netz R, Noel W (2007) The Archimedes Codex. Revealing the Secrets of the World’s Greatest Palimpsest, Weidenfeld & Nicholson, Great Britain. Mairan JJ d’Ortous de (1724) “Instruction abregée et méthode pour le jaugeage des navires” (Abridged instruction and method for admeasuring ships), in Hist. et Mem. de l’Acad. des Sci. 227-240 Mikhailov GK (1983) Leonhard Euler and the Development of Theoretical Hydraulics in the 2nd Quarter of the 18th Century, in German. 229-241, Burckhardt et al. Nowacki H (2002) Archimedes and Ship Stability, Proc. Intl. Multi-Conf./Euroconf. on Passenger Ship Design, Operation and Safety, Anissaras/Crete, edited by A. Papanikolaou and K. Spyrou, NTU Athens (Oct. 2001). Republ as Preprint No. 198, Max Planck Inst for the Hist of Sci, Berlin. Stevin S (1586) The Principal Works of Simon Stevin, 5 vols, ed. E.J. Dijksterhuis, C.V. Swets and Zeitlinger, Amsterdam (1955). In Vol I (General Introd, Mech): De Beghinselen des Waterwichts (Elements of Hydrostatics) and in the supplement to Beghinselen der Weegkonst (Elements of the Art of Weighing) the short note: Van de Vlietende Topswaerheit (On the Floating Top-Heaviness), first publ in Dutch. Truesdell C (1954) Rational Fluid Mechanics, 1687-1765, in Euler’s Collected Works, Series II, vol. 12, pp. VII-CXXV, issued by the Euler Comm of the Swiss Acad of Natural Sci, Lausanne. Truesdell C (1983) Euler’s Contributions to the Theory of Ships and Mechanics, an essay review, in Centaurus, vol 26, 323-335. Witsen N (1994) Aeloude en hedendaegsche scheeps-bouw en bestier (Ancient and modern shipbuilding and handling), Casparus Commelijn; Broer en Jan Appelaer Amsterdam (1671). Facsimile by Canaletto-Alphen Aan Den Rijn, Amsterdam (1979). Reprinted with commentary in Hoving, A.: Nicolaes Witsens Scheeps-Bouw-Konst open gestelt, (Nicolaes Witsen’s System of Ship Building), Uitgeverij Van Wijnen, Franeker.
  • M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, 181 Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_9, © Springer Science+Business Media B.V. 2011 The Effect of Coupled Heave/Heave Velocity or Sway/Sway Velocity Initial Conditions on Capsize Modeling Leigh S. McCue* and Armin W. Troesch** * erospace and Ocean Engineering, VirginiaA Tech **Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, Michigan Abstract Obar et al. (2001) described a series of model tests and a numerical model of a rectangular barge in regular beam seas with three degrees of freedom in roll, heave, and sway used to investigate the onset of capsize. The model had minimal freeboard resulting in significant water-on-deck. This paper employs the numerical model to examine specifically the impact of the coupling in heave with heave velocity initial conditions or in sway with sway velocity initial conditions on ultimate stability. Direct comparison with experimental results was used to validate the numerical model and can be found in Lee et al (Lee, 2001; Lee et al, 2006). 1 Introduction One of the greatest threats to vessel safety is capsize. While capsize stability has been experimentally studied in great detail for one degree of freedom, see for example (Soliman and Thompson, 1991; Thompson, 1997), at the time of the 2003 Ship Stability conference, little had been done with multiple degree of freedom experiments in which great attention was paid to initial conditions. This work extends the experimentation conducted by Obar et al. (2001) and the numerical simulation and experimental structure developed by Lee et al. (Lee, 2001; Lee et al, 2006) to the case of coupled state variables. Vassalos and Spyrou (1990) demonstrated that combinations of effects (in their case the effect of directional instabilities on transverse stability) can dramatically alter the ultimate stability of a vehicle. Thompson and de Souza (1996) show the dependence of stability on nonlinear coupling between roll and heave. While Soliman and Thompson (1991) and Thompson (1997) as well as the work of Taylan (2003)
  • L.S. McCue and A.W. Troesch182 thoroughly investigate the one degree of freedom problem, Vassalos and Spyrou (1990) and Thompson and de Souza (1996) make clear the importance of studying combined effects. Murashige and Aihara (1998) investigate the case of a forced flooded ship experimentally and mathematically. While they improve upon the traditional one- dimensional model through inclusion of the effects of water on deck, their model is based upon assumptions that cause one to neglect sway and heave degrees of freedom. The work presented in this paper seeks to expand upon the one degree of freedom models previously discussed, and the computationally intensive models developed for fully nonlinear simulation and/or water on deck models which are currently being developed and validated (see for example Huang et al, 1999 and Belenky et al, 2002). This paper discusses results from a quasi-nonlinear time domain model developed by Lee (2001) which is far less computationally intensive than its fully nonlinear counterparts allowing for investigation of far greater quantities of data. Additionally, the numerical model is validated by three degree of freedom experimental results conducted at the University of Michigan Marine Hydrodynamics Laboratory (Obar et al, 2001; Lee, 2001, Lee et al, 2006). Over one hundred eighty separate experiments were used in the study to validate the numerical model (Obar et al, 2001; Lee, 2001, Lee et al, 2006). For each test case a simple box barge was excited at a frequency of 6.8 rad/s. The waves acting upon the model had a wave height to wave length ratio of 0.02. Each release, however, featured different initial conditions in roll, roll velocity, sway, sway velocity, heave, and heave velocity. These values were measured in 1/30th of a second increments for the duration of the experimental run. Approximately 37% of the test runs resulted in capsize suggesting that initial conditions were a significant contributor to subsequent dynamics. While roll initial conditions (i.e. initial roll angle and roll velocity) are well known to influence capsize (e.g. Thompson, 1997), little has been said about heave initial conditions. A reduced- model nonlinear time-domain computer program is used in this work to provide guidance in interpreting the experimental results. Qualitative behavior of the capsize dynamics is presented in a number of formats. The roll phase space is characterized by “safe/unsafe basins” (e.g. Thompson, 1997). These figures show capsize boundaries based upon roll initial conditions. The evolution or change of these boundaries is investigated for varying sets of heave and sway initial conditions. From similar analyses, “integrity curves” are generated for systematic variations in the initial heave displacement and heave velocity as well as sway displacement and velocity. Each point on an “integrity curve” represents the ratio of safe (i.e. non-capsize) basin area for a given wave amplitude to a similar safe basin area derived from zero wave amplitude. In this way, the integrity curves show the relative influence of incident wave excitation on capsize relative to vessel safety in the absence of incident waves. Results from the simulation are presented here with discussions of accuracy and time of computation. Qualitative guidelines are suggested and recommendations are
  • Sway/Sway Velocity Initial Conditions on Capsize Modeling 183 made for determining whether experimental programs may be sensitive to test initial conditions. 2 Numeral Analysis 2.1 Definition of Numerical Model The works of Lee (2001) and Lee et al (2006) develop a quasi-nonlinear three degree of freedom ‘blended’ hydrodynamic model to simulate highly nonlinear roll motion of a box barge. The model uses an effective gravitational field to account for the centrifugal forces due to the circular water particle motion in addition to the earth’s gravitational field. This results in a local time dependent gravitational field normal to the local water surface. Additionally the model assumes long waves. Equation 1 below gives the equations of motion for the numerical model in global coordinates where a and b are added mass and damping coefficients, m and I are the mass and moment of inertia of the body, ge is the time dependent effective gravitation, fD are diffraction forces, is the time dependent displaced volume of water, and GZ is the ship’s time dependent roll righting arm.                                                                                D e D e D e gg g g g g g g g g cg fGZg fmgg fg y x b y x bb b bb y x aIa am aam 4 3 2 2142 33 2422 4442 33 2422 4 3 2 00 000 000 0 00 0 0 00 0      (1) The global and body fixed coordinate systems for the model are defined in Figure 1.
  • L.S. McCue and A.W. Troesch184  Fig. 1 Coordinate system definition, Model scale definitions: 18.25 cm draft, 1.12 cm freeboard, 66 cm length, and 30.48 cm beam. While the numerical model cannot account for the dynamics of water on deck, such as sloshing, it does account for hydrostatic effects due to deck immergence. The forcing due to the waves is modeled as a cosine wave. To simulate laboratory transients, the wave “ramps” from zero until steady state. An experimentally determined envelope curve was generated to create a ramped cosine wave that closely resembles the waves generated in the tank (Obar et al, 2001). All release times for the model are defined relative to the maximum transient wave crest as shown in Figure 2 below and indicated by the notation “to”. Fig. 2 Sample wave profile (top) with enlarged region showing definition of points relative to to (bottom).
  • Sway/Sway Velocity Initial Conditions on Capsize Modeling 185 While this ‘blended’ model has admitted weaknesses due to the simplifications employed, it has the distinct advantage of computational efficiency. Other, multi- degree of freedom models that simulate various dynamics of water on deck run significantly slower than real time (Belenky et al, 2002). However, this model, whose results are qualitatively experimentally validated (Lee et al, 2006), runs in a fraction of the actual run time. The work that makes up this numerical study represents years worth of real time data; using this quasi-nonlinear model such data can be collected in a matter of days. 2.2 Numerical results As discussed in the work of Soliman and Thompson (1991), one can consider a set of model releases in which all initial conditions, save roll and roll velocity, are held constant. A sweep is made of roll and roll velocity initial condition pairs with conditions resulting in capsize marked in black and non capsize left white yielding a ‘safe basin’. Figure 3 shows a sample safe basin generated with this numerical model (note e denotes excitation frequency, n represents vessel natural frequency in roll, similarly te and tn represent excitation and natural periods respectively). Fig. 3 Safe basin for release time of to-2 with sway, sway velocity, heave, and heave velocity initial conditions set equal to zero at release. en≈3, h/=0.019.
  • L.S. McCue and A.W. Troesch186 Fig. 4 Safe basin for release times starting from to-2 in 30 degree increments with en≈3, h/=0.019, roll angle from -30 to 30 deg and roll velocity from -30 to 30 deg/s; i.e. the left hand column represents safe basins at to-2, to-1.5, to-1, to-.5, to, to+.5, to+1, and to+1.5. All heave and sway displacements and velocities initially set equal to zero. x and y axes represent roll and roll velocity respectively.
  • Sway/Sway Velocity Initial Conditions on Capsize Modeling 187 One can then consider a set of safe basins with some third parameter varied. For example, Figure 4 shows a series of safe basins for different release times. The figure shows safe basins for 48 different release times starting from to-2 until to+2. Thus a shift between two safe basins represents a 30 degree shift in phase location along the wave profile (i.e. 1/12 ). This figure illustrates how crucial accurate accounting of time, and implicitly sway location, is on predicting vessel stability. Additionally, this plot demonstrates the impact transients in the incident wave profile have upon the safe basins. The safe basins before steady state, such as from to-2 to to-1, exhibit no periodicity (i.e. to-2 and to-1’s safe basins are quite different), while basins from to to to+2 produce a periodicity in that every 360 degrees of phase (or one wave length) the safe basin is virtually identical (i.e. to+.5 and to+1.5 are nearly the same). It should be noted that a change in sway displacement can be explicitly written as a shift in time. For example, upon reaching steady state, a sway displacement equal to one wave length at time tx could instead correspond to a sway displacement of zero at a time tx+1, that is time moved forward through one wave period. Thus a shift in time corresponds to a change in sway displacement if time is referred to by a particular location in the wave profile, e.g. the maximum wave crest, to (Figure 2). As described by Thompson (1997), one can then introduce the idea of integrity values. Integrity is a ratio of safe area for given initial conditions to safe area for some reference case. For a typical integrity curve, wave frequency and all non-roll initial conditions would be held constant while wave amplitude is varied. As an example, Figure 5 presents an integrity curve for the box barge numerical model described in the previous section. In Figure 5 sway, sway velocity, heave, and heave velocity are all set to zero at the instant of release. A safe basin is generated for varying roll and roll velocity initial conditions (between -20 to 20 degrees and –15 to 15 degrees/sec respectively). This is done for multiple wave amplitudes with integrity values being calculated as the ratio of safe area at a particular wave height to safe area for a wave height of zero. Thus the curve begins at one. The resulting curve is an integrity curve as defined by Thompson (1997) in which one notes the crucial characteristic ‘Dover cliff’ (1997) at which point there is a distinct, rapid loss of stability. One can note that in Figure 5, integrity values are greater than one for wave amplitude to wave length ratios of 0 to approximately 0.007. Physically, integrity values greater than 1.0 indicate that the presence of waves at certain wave amplitudes can have a stabilizing effect upon the vessel relative to the zero wave condition.
  • L.S. McCue and A.W. Troesch188 0 0.2 0.4 0.6 0.8 1 1.2 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 Wave Amplitude per Wave Length (o/) Integrity Fig. 5 Integrity Curve. Sway, sway velocity, heave, and heave velocity 0 at time of release. Te/Tn~3. In Lee et al (2006) a detailed study is conducted considering the ways in which these curves shift for varying initial conditions of sway, sway velocity, heave, and heave velocity. This results in series of curves such as those found in Figure 6. In this case one sees the variations in the location of the cliff; however, direct comparison is difficult as the normalization factor for each curve varies. This is analysed in detail for the 6 state variables in Lee et al (2006). Fig. 6 Heave Integrity Curves. Sway, sway velocity, and heave velocity 0 at time of release. Te/Tn~3. Note significant change with respect to heave in location of cliff on curve but not in x- axis intercept (location where integrity is zero). Heave (normalized by beam) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 Wave Amplitude per Wave Length ( /) Integrity -0.09 -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 -0.09 0.09
  • Sway/Sway Velocity Initial Conditions on Capsize Modeling 189 While this lends great insight into anticipated behaviour from individual state variable variations, it still cannot fully capture the idiosyncrasies of experiments as there is no graphic coupling amongst the remaining state variables, sway, sway velocity, heave, and heave velocity. Therefore, instead of viewing the parameters as integrity curves, one can consider a surface, in which integrity (defined on the roll and roll velocity phase space plane) for a fixed wave amplitude is calculated for pairings of two other state variables such as sway and sway velocity (Figure 7) or heave and heave velocity (Figure 8). Each of these surfaces is for a fixed wave height and frequency (h/=.019 and en≈3) and is normalized such that the integrity value is 1 when sway, sway velocity, heave, and heave velocity are zero. Figure 7 presents a three dimensional integrity surface where each point is an integrity value representing the ratio of safe area to unsafe area for a roll/roll velocity grid as a function of sway and sway velocity. Heave and heave velocity are defined as zero at the time of release. Additionally excitation frequency and incident wave amplitude are constants. This figure clearly shows the relative independence of stability on sway velocity initial conditions. If one were to pick a fixed value of sway the plot collapses to a two-dimensional curve representing integrity as a function of sway velocity. This curve is relatively horizontal and constant thus showing the independence of integrity on sway velocity. For example, for sway equal to one wave length (or correspondingly zero wave length), the maximum integrity value as a function of sway velocity is 1.24 and the minimum is 0.70. While this is non-trivial, comparatively speaking, this is a relatively small change in contrast to the effects seen by changes in sway displacement, heave displacement, and heave velocity. If one were to instead consider a constant sway velocity, we see integrity values as a function of sway vary from zero to 2.2. Therefore one can see that for any fixed sway velocity value, there is a strong dependence of the integrity value upon sway position. In fact, for values of sway from approximately .15 to .65the integrity values are zero irregardless of sway velocity. This indicates for some initial release positions on a wave (the region 15%-65% of a wave length beyond a peak) the vessel shall always capsize regardless of initial roll angle, initial roll velocity, or sway velocity for a wave amplitude to wave length ratio of 0.0096 and an excitation frequency to natural frequency ratio of approximately 3. This is due to the fact that this wave amplitude is in a particularly steep region of the integrity curve. Referring to Figure 5, one should note the steep drop off at approximately o/=.01. For a set of sway integrity curves the location of this ‘cliff’ is relatively invariant. More details on this are given by Lee et al (2006). Figure 8 demonstrates the intricate interdependence between heave and heave velocity state variables in determining ultimate vessel stability. For the surface presented in Figure 8 sway and sway velocity are zero at the time of release and
  • L.S. McCue and A.W. Troesch190 wave height and frequency are constants for each point determining the surface. It should be noted though that after the time of release the vessel is unconstrained in all three degrees of freedom, thus sway and sway velocity are allowed to evolve in time just as heave, heave velocity, roll and roll velocity evolve. This is true for the data comprising both Figures 7 and 8. Unlike Figure 7 we note there are no similar statements we can make regarding a relative independence. Both heave and heave velocity have substantial influence on the ultimate state of the vessel and a coupling between the two conditions of heave and heave velocity can significantly alter the location of the vessel on such an integrity surface indicating a dramatic shift in vessel safety. Only generally can we say that maximum integrity values occur when heave velocity initial conditions are near zero for all heave. 3 Conclusions This paper presents the results of a thorough computational study into the inter- dependence of initial conditions in multiple degrees of freedom on predicting ultimate state of a simple box barge. While the hydrodynamic model is highly simplified it serves as a useful tool having demonstrated the influence coupled degrees of freedom can have on capsize prediction. The final two figures, 7 and 8, demonstrate the dependence of vessel stability on heave, heave velocity, sway and to a lesser extent sway velocity. While many models used in the present day literature focus upon one degree of freedom or a reduced order model, this work has shown that such models can be vulnerable to neglecting crucial dynamics. It is not sufficient to consider simply one state variable such as roll angle nor two state variables models coupling roll with roll velocity. This work demonstrates that elaborate couplings of multiple state variables in roll, sway and heave can significantly alter predictions of safe versus capsize and thus one must exercise care in the use of reduced order models. Acknowledgements The authors would like to acknowledge the support of the National Defense Science and Engineering Graduate Fellowship program for sponsoring this research. Additionally the authors would like to thank Young-Woo Lee and Michael Obar whose work laid the basis for this paper.
  • Sway/Sway Velocity Initial Conditions on Capsize Modeling 191 Fig. 7 Sway/sway velocity integrity surface. Heave and heave velocity 0 at time of release, to. en≈3, h/=0.019. Fig. 8 Heave/heave velocity integrity surface. Sway and sway velocity 0 at time of release, to. en≈3, h/=0.019.
  • L.S. McCue and A.W. Troesch192 References Belenky VD, Liut KW, Shin YS. Nonlinear ship roll simulation with water-on-deck. Proc of the 2002 Stab Workshop. Webb Inst. 2002 Huang J, Cong L, Grochowalski S, Hsiung CC. Capsize analysis for ships with water shipping on and off the deck. Twenty-Second Symp on Naval Hydrodyn. Washington D.C. 1999 Lee YW. Nonlinear ship motion models to predict capsize in regular beam seas. Doctoral dissertation, Dep of Naval Archit and Mar Eng. University of Michigan Ann Arbor MI 2001 Lee YW, McCue LS, Obar MS, Troesch AW. Experimental and numerical investigation into the effects of initial conditions on a three degree of freedom capsize model. J of Ship Res 2006 50:1 63-84 Murashige S, Aihara K. Coexistence of periodic roll motion and chaotic one in a forced flooded ship. Int J of Bifurcation and Chaos 8:3 1998 619-626 Obar M, Lee YW, Troesch A. An experimental investigation into the effects initial conditions and water on deck have on a three degree of freedom capsize model. Proc of the Fifth Int Workshop on the Stab and Oper Saf of Ships. September 2001 Soliman, Thompson. Transient and steady state analysis of capsize phenomena. Appl Ocean Res 1991 Taylan M. Static and dynamic aspects of a capsize phenomenon. Ocean Eng. 30: 2003 331-350 Thompson J. Designing against capsize in beam seas: Recent advances and new insights. Appl Mech Rev. 50:5 May 1997 Thompson JMT, de Souza JR. Suppression of escape by resonant modal interactions: in shell vibration and heave-roll capsize. Proc of the Royal Soc of London. A452 1996 2527-2550 Vassalos D, K Spyrou. An investigation into the combined effects of transverse and directional stabilities on vessel safety. Fourth Int Symp on the Stab of Ships and Ocean Veh. Naples, Italy. 519-527 1990
  • M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, 193 Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_10, © Springer Science+Business Media B.V. 2011 The Use of Energy Build Up to Identify the Most Critical Heeling Axis Direction for Stability Calculations for Floating Offshore Structures Joost van Santen GustoMSC the Netherlands Abstract For offshore structures like semi submersibles and jack-ups, hydrostatic stability is to be determined for what is called the weakest axis, which is not necessarily the same as the longitudinal axis of symmetry of the structure. When allowing trim to take place, the determination of the critical axis is complicated as free trimming leads to multiple solutions regarding the position for a given heel angle. It will be shown that for a freely floating structure, looking at the increase in potential energy can be used to identify those axis directions which are critical as well as realistic. The theoretical results will be illustrated with detailed data obtained for a two typical offshore structures using a standard stability program. Nomenclature  = displacement (t)  = heel angle  = axis direction (twist angle)  = trim angle Cxy = cross moment of the waterline plane for rotation around its centre ≈∫x.y.dA CoB = Centre of Buoyancy CoG = Centre of gravity E = potential energy divided by displacement GML = metacentric height for trim at a particular heel angle
  • J. van Santen194 GMT = metacentric height for heel at a particular heel angle Mx = moment around the longitudinal horizontal axis divided by dis- placement My = moment around the transverse horizontal y axis divided by dis- placement VCB = vertical centre of buoyancy above the centre of gravity x = horizontal axis along the initial heeling axis, forward positive xcob, ycob = location centre of buoyancy relative to CoG y = horizontal axis perpendicular to the x axis (right handed system) 1 Introduction Historically, determination of the stability offshore rigs is seen as extension of stability for ships. For ships the longitudinal axis is taken as the heeling axis, where trim is allowed to remove the trimming moment. Early on, the offshore industry has recognized that the most critical axis is not necessarily the longitudinal axis. So, the wording critical axis was introduced, but as a kind of heritage one also had to consider free trim. This directly leads to a problem. This problem is that a given position can be defined by infinite combinations of axis directions, heel angle and trim. Two angles suffice to uniquely define the position. Using three angles means that one is superfluous. In the 80’s several papers appeared introducing the use of pressure integration to replace the conventional way of slicing up the structure and calculation of the contribution of each slice, (Witz and Patel, 1985, van Santen, 1986). In the publication of van Santen (1986), the problems mentioned above were raised and a way to avoid them by using free twist was introduced. In this paper, also the concept of energy build up was touched upon. More recently (Breuer and Sjölund, 2006), the problem was looked at again and using the build up of energy was proposed as a solution. In the underlying publication, the increase in potential energy of a structure due to forced heel with and without trim will be looked at and examples will be shown. This paper deals with freely floating structures. Effects due to mooring or dynamic positioning are specifically excluded.
  • Axis Direction for Stability Calculations for Floating Offshore Structures 195 2 Example with a Barge For a rectangular barge (Figure 1) with a raised forecastle, we can construct the righting arm curves for a range of axis directions (without trim) as shown in Figure 2. Fig. 1 Barge dimensions Fig. 2 Righting arm curves, no trim Applying free trim has dramatic consequences as is shown in Figure 3. For some axis directions the righting arm curves stop prematurely. Why is this? In order to analyze this, the trimming moment as a function of trim angle is to be studied, see VCG Seawater density 1025 kg/m3 Intact draft 5.0 m TCG 140.0 m8.5 m 36.0 m 8.0 m 25.0 m LCG Volume 25200 m3 70.0 m 0.0 m 17.0 m 40 35 30 25 20 15 10 5 0 302010 Heel angle (deg) 0 0 40 60 80 90 Arm(m)
  • J. van Santen196 moment is zero is the free trim angle belonging to a particular heeled situation. Fig. 3 Righting arm curves, free trim Fig. 4 Trimming arm curves For an axis direction of 80 deg, when heeled beyond 5.6 deg heel, the zero crossings disappear, meaning that there are no solutions with zero trim moment. When passing a certain position, continuation of the righting arm curve can only Trimming arm for an axis direction of 80 deg zero value indicates a free trim position No solution Heel angle Trim angle (deg) Unstable solution -50 -40 -30 -20 -10 0 7 6 3 0 -3 7° 6° 5° 4° 3° 6 5 4 3 Trimmingarm(m) Stable solution Figure 4. The trim angle for which this
  • Axis Direction for Stability Calculations for Floating Offshore Structures 197 be achieved by reducing the heel whilst at the same time increasing the trim angle. This leads to the righting arm curve as shown in Figure 5 and the trim angle as shown in Figure 6. Fig. 5 Continuous righting arm curves, free trim Fig. 6 Trim angle for continuous heel For intact offshore rigs, some authorities require a range of positive stability of 30 (or 36) deg. For a longitudinal axis direction this criterion would be met, but for an axis direction of 80 deg, the structure would fail to meet this criterion. So, one could say that 80 deg axis direction is the most critical one and the rig would fail to comply with the range requirement. Continuous path 15 10 5 Rightingarm 0 -5 -10 -15 Stable in trim Unstable in trim Heel angle (deg) 0 1 2 3 4 5 6 -15 -10 -5 Trimangle[deg] -20 -25 -30 -35 Heel angle (deg) 0 0 1 2 3 4 5 6 Continuous path
  • J. van Santen198 Looking at the structure (figure 1) it is obvious that one should select an almost longitudinal axis as the most critical axis direction and not 80 deg. But for a more generic structure it is not that easy to identify the critical axis direction. Why is it that from visual observations we reject 80 deg and accept 0 deg ? The key can be found in the rate at which the potential energy in the system increases with heel. For 0 deg axis direction this is far less than for 80 deg. The potential energy is given by the negative of the vertical distance (VCB) between the centre of gravity CoG and the centre of buoyancy CoB. Energy VCB displacement,   (1) At the equilibrium position the energy is taken as the reference value, E0. For a change in heel, the increase in energy is given as (2) The input of energy is the work done by the overturning heeling moment given by (3) ) is the overturning moment divided by displacement. This results in: (4) Note that for a free floating structure, a positive overturning moment results in a positive righting arm, but this means in fact that the counteracting restoring moment given by ycob is negative. The most critical axis can be viewed as the axis direction for which a given heel angle is reached with the least effort. In this way, the least energy is to be fed into the system in order to reach that particular heel angle. Figure 7 shows the amount of energy fed into the barge depending on axis direction and heel angle.     0 0 E E E VCB VCB          0 M d           0 0 0 VCB VCB M d righting arm d               where E will be used to denote energy divided by displacement, so E = -VCB. where M(
  • Axis Direction for Stability Calculations for Floating Offshore Structures 199 Fig. 7 Energy surface for the barge The trim is fixed to zero. From this figure it is seen that for an axis direction of about 0 deg the lowest amount of energy is needed to reach a given heel angle. This agrees with the subjective feeling that the most critical axis is almost longitudinal. The problem now is how to determine the axis direction for which the energy increase is lowest. But before we do so, another example will be given, as found in the publication by Breuer and Sjöland (2006), see also Figure 8. 3 ABS Jack-Up Following discussions between MSC and ABS on the approval of a particular MSC jack-up, related to the Range of Stability (RoS) requirement (ABS rules 2008), Breuer and Sjölund (2006) published their paper in which they showed energy contour plots for a range of trim and heel angles. For the damage as indicated in Figure 8, the increase in energy depending on axis direction and heel angle is given in rate of increase is found for an axis direction of about 320 deg. Figure 9. This figure shows that the lowest
  • J. van Santen200 Fig.8 Dimensions of the studied jack Fig. 9 Energy surface of the studied jack up, damaged condition For this direction, the righting arm is shown in Figure 10 (free trim). Selecting another axis direction (280 deg) results in a vanishing righting arm curve, as shown in Figure 11. Note the large trim angles related to the vanishing arm curve. Using an axis direction of 280 deg may lead to the false impression that the RoS criterion is not. Actually it is met, but it is for an axis direction of 320 deg.
  • Axis Direction for Stability Calculations for Floating Offshore Structures 201 Fig. 10 Righting arm and trim angle, axis direction 320 deg Fig. 11 Righting arm and trim angle, axis direction 280 deg 4 Axis Convention When using heel, trim and twist, it is important to have a clear understanding of their meaning. For the remainder of this publication, the following convention is used when positioning a structure in a heeled condition (see Figure 12): Righting arm (m) 2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 Trim angle (deg) RoS=7+1.5*2.6 deg=10.9 deg Heel angle 0 5 10 15 Righting arm (m) 20 15 10 5 0 0 2 4 6 8 10 -5 Trim angle (deg) Heel angle
  • J. van Santen202  first the heeling axis direction is chosen,  the structure is heeled around this axis  the structure is trimmed around a transverse axis which was initially horizontal but changes direction due to heel. 5 Free Trim Versus Free Twist As is seen above, looking at energy increase during heel is a clear indicator for the determination of the critical axis direction. The next question is how to deal with trim. For ships, introducing free trim is clearly meant to obtain the lowest righting arm. This is easily seen when going from the free trim to the situation with nil trim. When going from the trimmed to the free trim situation, energy is dissipated, thus the righting arm is less than for the fixed axis direction and zero trim situation. The question is if this also works when varying the axis direction. For this purpose consider a heeled position with both a zero trimming moment and a zero moment around the twist axis which is the initially vertical axis. When applying a small change in the trim angle (d) and in the axis direction (d) reactive moments will result. For a constant displacement, using conventional hydrostatic considerations, these moments can be determined by looking at the horizontal and vertical components of the rotations. The horizontal component () causes a change in waterline shape and in CoB position. The combined effect translates into a reactive moment -GML . The rotation () around the vertical axis causes a transverse movement of the CoB which causes a reactive moment ycob . Thus, for a change of trim, d, the externally imposed moment needed for this change in trim becomes:
  • Axis Direction for Stability Calculations for Floating Offshore Structures 203 Fig. 12 Application of twist, heel and trim Select axis direction axis system through CoG Shift to CoG input axis system Axis direction Heel angle Heel angle Heel angle My Mx Position after structure is heeled around a horizontal axis Position after structure is trimmed around an inclined transverse axis Trim angle x y z waterline x y z x y z
  • J. van Santen204  t L cobMy GM cos y sin d    (5) Note that the value of GM is the instantaneous metacentric height for the given heeled position. Similarly, due to a change of axis direction d the externally imposed moment becomes:  a L cobMy GM sin y cos d     (6) These moments are around a horizontal axis. When considering energy input, the moment is to be taken along the axis around which the rotation takes place. For instance, for trim, the moment around the trim axis is Myacos. Thus, for trim the energy input starting from the free twist, zero trim, position is:   2 t L cob 1 dE cos GM cos y sin d 2      (7) Due to a change in axis direction the energy input starting from free twist is:   2 a L cob 1 dE sin GM sin y cos d 2      (8) The term between the brackets in Equation 8, for small heel angles the term between the brackets can be approximated by GML  - GMT  cos. So, for small heel angles, it is in general small but also positive. For larger angles it is in general positive. So, in general, when starting from the free twist position with zero trim and applying either a small change in twist or in trim an increase in energy in the system is found. Thus, the free twist position is the position with the lowest potential energy for a given heel angle. The question arises if there is a reduction in energy when going from a free twist position with zero trim to a new position (with a change in twist of d) and letting the rig free to trim with angle d. The energy input needed to twist is given by dEa. By letting the rig free to trim, the energy recovered is – dEt. So the total energy input is dEa - dEt. The twist angle is obtained by imposing a moment on the rig. This moment is nullified by letting it free to trim. So: (7) is in general positive. In equation
  • Axis Direction for Stability Calculations for Floating Offshore Structures 205 a tMy My  (9) Using Equations (5) and (6) gives the relation between d and d: L cob L cob GM cos y sin d d GM sin y cos         (10) By substituting Equation (10) in Equation (8) the difference in energy input is found to be a t 2 L cob cob L cob dE dE dE GM cos y sin1 d y 2 GM sin y cos          (11) Generally, the following applies:  GML is positive,  ycob is negative for a positive righting arm Thus, for the weakest axis, the numerator GML cos -ycob sin is in general positive. The denominator is also in general positive. So, when going from a free twist to a free trim situation the energy input is positive. This leads to the conclusion that the free twist situation contains less energy than fixing the axis direction combined with free trim. The free twist position equals the situation with a local minimum energy as both moments Myt and Mya are zero. For the jack-up shown in Figure 8, the numerical values for small variations in trim and twist around the free twist point are shown in Figure 13. These are based on the calculations with a stability program as well as on the approximations given above. Also, the calculated relation according to for which energy Ea is exchanged with Et is shown (“lowest envelope”). This figure shows the validity of the theoretical formulations. It is important to be aware that in the free twist method, trim is always zero. Hence, the heel angle is always equal to the steepest slope of an initial horizontal plane (like a deck). Equation (10) between axis and trim
  • J. van Santen206 Fig. 13 Energy depending on trim 6 Change in Axis Direction Due to a Change in Heel Angle In section 5 it is shown that the free twist method results in the lowest amount of potential energy build up for a given heel angle. For a free twist situation, the moment around the initial vertical axis is zero. When making a small step in the heel angle, the change in axis direction can be estimated using this condition. For a constant displacement a heel increase d results in moment around the twist axis of -Cxy · d, where Cxy = cross moment of inertia of the waterline for a rotation around its centre of floatation. Making this equal but opposite to the moment due to a small change in twist (Mya, quation (12) It is seen that the axis direction is influenced by both the metacentric height (GML) and the righting arm (which equals -ycob). xy L cob C / volume d d GM sin y cos        Energy depending on trim for varying axis directions damaged rig, 10 deg heel 0.005 0.004 0.003 0.002 0.001 SE-10 -0.001 0 0.5 Approximation axis 324,3 deg Calculated lowest envelope -0.5-1 1 326,3 325,3 324,3 323,3 322,3 321,3 energy(m) trim angle (deg) E (6)) the change in axis direction results
  • Axis Direction for Stability Calculations for Floating Offshore Structures 207 7 Stability of the Solution The free twist position is characterized by the twist moment Mya being zero. It is not necessarily a position with the lowest energy, but it can also be a position with the highest energy. For a given heel angle, there are several axis directions for which the energy is either minimum or maximum. The maximum is by definition an unstable position, the minimum is a stable position. Stability is indicated by the energy input Equations (8) (for twist) and (7) (for trim). If the term GMLsin+ycobcos For an intact structure at a small heel angle, the following approximations apply: ycob ~ -GMT  sin() ~  cos() ~ 1.0    L T stability term 0 means stable GM GM    (13) Clearly, for small angles the position is stable when heeling is around the axis with the smallest GM value. For larger angles, the actual values of GML and ycob are to be considered. When the structure is in free to trim, also the stability for trim should be looked at. For trim to be unstable, the term GMLcos-ycobsin is to be negative. Keeping in mind that -ycob is the righting arm, it is seen that for a positive righting arm this is the case as long as GML, being the metacentric height for the given heeled position, is positive. 8 Case Studies For two structures, analyses have been done into the increase in potential energy during progressive heeling. These structures are:  ABS jack-up,  a simplified semi submersible 8.1 ABS Jack-Up Intact. ucture, see Fig damaged condition. For the intact condition, Figure 14 shows the energy surface for a range of heel angles (0-24 deg) and a range of axis directions (0-360 deg). is negative, the position is unstable in twist. This ure 8 has been analyzed for both the intact andstr
  • J. van Santen208 Fig.14 Energy surface intact It is seen that for larger heel angles 3 axis directions are present following the local minimum path. These are 900 , 2100 and 3300 . The 210 and 330 deg directions are in fact identical due to the symmetry of the structure. From the graph, it is also seen that the 330 (or 210) degree yields the lowest energy increase. For angles exceeding about 6 degrees heel there are 3 maxima and 3 minima in the plot. What is not clear is that for smaller heel angles there are only 2 maxima and 2 minima. The extremes at around 35 deg and 145 deg disappear for small heel angles. This is shown in Figure 15 which shows the axis directions which have a zero twist moment depending on the heel angle. Fig. 15 Axis directions for minimum energy build up, intact
  • Axis Direction for Stability Calculations for Floating Offshore Structures 209 Damaged. The damage case has a damaged compartment as indicated in Figure 8. For a VCG of 23.45 m, the range of stability criterion of ABS is just met. This range should be 7 deg + 1.5 x steady heel (2.59 deg) = 10.89 deg. The plot of the energy (Figure 9) shows a minimum value for an axis direction of around 320 deg. Figure 16 shows the axis direction for which free twist is satisfied, i.e. where the moment around the y axis is indeed zero. The figure also shows the estimated axis direction based on Equation (12). In this estimate, the start value at zero heel is taken from the calculations with the stability program. The other values are based on the summed increments. A very good fit is seen between the estimated and the actual axis directions. For the damaged case, at larger heel angles, there are two paths following a local extreme in the energy surface. For small heel angles there is only a single path. Fig. 16 Axis directions for minimum energy build up, damaged 8.2 Semi-Submersible Figure 17 shows a semi-submersible which at the given displacement has an intact draft of 9.106 m. Both the intact and damage condition pose a challenge when calculating the stability. Intact. Figure 18 shows the energy surface plot for the intact condition. Note that because of symmetry, the data for an axis direction of α is also found for axis direction α + 180. It is seen that for moderate heel angles, the minimum energy is 18 16 14 12 10 8 6 4 2 0 200 250 Estimated path (eq13) Minimum energy path 300 Axis direction (deg) Heelangle(deg) 350 Stable Unstable
  • J. van Santen210 found for an axis direction of 180 deg (and 0 deg). This is even better seen in the righting arm plot, Figure 19. When for 180 deg axis direction, heel is increased beyond 25.5 deg, the path follows a local unstable maximum instead of a local stable minimum. Instead, for larger heel angles, the minimum energy is found for axis directions of about 135 and 225 deg. A more detailed analysis can be done by looking at the twist moment depending on heel and axis direction. Fig. 17 Shape of the semi submersible Fig. 18 Energy surface, intact
  • Axis Direction for Stability Calculations for Floating Offshore Structures 211 Fig. 19 Surface plot of righting arm Fig. 20 Surface plot of the trimming arm Figure 20 shows this as a surface plot of xcob versus heel and axis direction. The free twist locations are those for which xcob equals 0. The combination of heel and axis directions for which the twist moment (or xcob) is zero are identified by the change in color in Figure 20. The stable and unstable positions can be identified by looking at the slope dxcob/ d for a given heel angle. Using this figure, the stable and unstable paths for which there is an extreme in the energy build up are as given in Figure 21 for axis directions between 120 and 240 degrees.
  • J. van Santen212 Fig. 21 Stable and unstable paths Starting with zero heel, a gradual increase in the overturning moment will result in the rig to follow a seemingly erratic path: heel axis direction 0 – 1.6 180 deg 1.6 – 4.6 151-154 deg or 209 – 206 deg 4.7 – 25.4 180 deg 25.5 –50 133-138 deg or 227 – 222 deg At hindsight, the unstable area for very small heel angles is also seen in Figure 19 by a barely noticeable hill for axis 180 and heel 4 deg. Damaged. When damaging the rig by removing the pontoon corner part as indicated in Figure 17, the energy surface looks more simple than for the intact rig. The axis direction for minimum energy increase is around 140 deg, see Figure 22. Still, also in this case, the axis direction changes considerably for increasing heel as shown in Figures 22 and 23 When actually increasing the heel angle in a stepwise manner, the rig will suddenly change position when passing the 12 and 23 degrees heel, see Figures 23 and 24. At 12 degrees heel, it will change from axis direction about 140 deg to about 166 degree. At 23 degree heel it will change back to about 130 deg axis direction. Further study showed that this behavior hardly depends on the VCG. 120 50 Stable branches Unstable branches 40 30 20 10 0 170 Axis direction (deg) Intact, VCG=40 m Heelangle(deg) 220
  • Axis Direction for Stability Calculations for Floating Offshore Structures 213 Fig 22 Energy plot, damaged Fig. 23 Relation between heel angle and axis direction, free twist 40 35 30 25 20 Heelangle(deg) Axis direction (deg) 15 10 5 120 130 140 150 160 Jump in axis direction for increasing heel Jump in axis direction for decreasing heel 170 180 0
  • J. van Santen214 Fig. 24 Righting arm following free twist 9 Use of Minimum Energy Path in Evaluating Stability Criteria When evaluating stability criteria, a distinction can be made between those without and with external influence. An example of the first group is ABS’s range of stability criterion for jack-ups. An example of the second group is the well known requirement on ratio between the area under the restoring moment and the wind overturning moment curves. For the first group, the minimum energy path can be followed as being the most critical, i.e. it requires the least effort to reach a particular heel angle. For the second group, both the magnitude of the restoring moment and of the overturning moment is to be considered. This highly complicates the task of selecting the most critical heeling axis direction. When following the minimum energy path, it is relatively easy to adapt the wind overturning moment to the instantaneous axis direction. But, this still assumes that the direction of the wind overturning moment is the same as that of the restoring moment. In general, it is well possible that this is not the case and that the wind overturning moment has a trimming component as well. But, this raises the question if one should absorb this by letting the unit trim or if the axis direction should be modified. It is also possible that the wind overturning moment for other directions is higher and thus more governing. Another way would be to look directly at the ratio between energy inputs versus energy build up for all axis directions without considering trim. When including possible downflooding, the complexity of the calculation increases further. 5 4.5 4 3.5 3 Rightingarm(m) Heel angle (deg) 2.5 2 1.5 1 0 10 20 Jump in righting arm 30 40 0.5 0
  • Axis Direction for Stability Calculations for Floating Offshore Structures 215 Apart from the calculation difficulties, the major issue is the nature of the calculation. Traditionally, the effect of wind for an otherwise calm sea is looked at. The effect of waves and resulting motions can be important, especially for a damaged structure (van Santen, 1999). Also, the calculation of the wind overturning effect is usually simplified in that the reactive force is assumed to work at the lateral center of resistance and that the forces due wind and the reactive forces do not introduce a yawing moment. Including the effect of mooring or DP makes the analysis even more complex. In view of this, there seems to be no justification to focus on one detail whilst ignoring other effects which may be much more important. In the end, we should not forget that the criteria are quite abstract. Being abstract, they lend themselves to consistency and reproducibility. The end results are not so much influenced by the details of the method used. As such, they fulfill the requirement of a criterion of which the evaluation is clearly defined and can be reproduced independently. By Couser (2003) the difficulty in programming the calculation of the AVCG curves as limited by the various requirements was mentioned. The examples shown here indicate that apart from the vagueness in the criteria, the structure itself can cause problems. For the semi-submersible case at transit draft it is almost impossible to do a proper AVCG calculation. A possible way out is to use a fixed axis direction with nil trim. For a righting arm curve constructed in this way to be acceptable, it should show only modest trimming moments as this indicates that the energy depletion by letting it free to twist or trim is small. 10 Conclusions 1. Extending the free trim approach as used for ships to offshore units, in combination with a varying heeling axis direction, may lead to severe interpretation problems. 2. A free trim or free twist approach is sensible as this generally leads to the slowest build up of potential energy for increasing heel angle. Thus the lowest righting arm curve is achieved. A pre-requisite is that the rig remains stable in trim and twist for the relevant range of heel angles. 3. The free twist approach with zero trim leads to the lowest gain in potential energy during heeling. This is based on both theory and data obtained from direct calculations. Thus when left free to trim and free to twist, the result is that the twist angle varies and that the trim angle remains zero. 4. Application of the lowest gain in energy approach is an unambiguous way to define the most critical or weakest axis. 5. For the determination of the critical axis direction, other effects are to be considered as well. When wind overturning is in the criterion, axes other than the weakest based on energy, may have to be considered. Also when looking at openings, other axis directions have to be looked at.
  • J. van Santen216 For these complex cases, it is strongly suggested to perform the calculations for any axis direction and with the trim fixed at zero. Postscript Since the original publication in 2009, some errors have been removed. Most important is the change in width of the compartment in figure 17 from 7 to 6 m. This has an important influence on the results. Also later at the City University conference 2009, an additional paper was published (Santen 2009) looking into more detail at the various methods including the Steepest Descent Method as proposed by Breuer and Sjölund, 2006. It explains and analyzes the SDM, gives details and shows results obtained with this method. The author performed some tests with scale models of the jack up and semisubmersible which confirmed the findings reported in this paper in a qualitative way. I owe many thanks to Andy Breuer for having many open minded and fruitfull discussions about the problems met when doing stability calculations for offshore structures. His encouragement to publish my views resulted in this and in the City University paper. References ABS rules for building and classing Mobile Offshore Drilling Units 2008. Breuer JA and Sjölund KG (2006) “Orthogonal Tipping in Conventional Offshore Stability Evaluations”, Proc of the 9 th Int conf on Stab of Ships and OceanVeh (STAB 2006), Rio de Janeiro. Santen JA (1986) “Stability calculations for jack-ups and semi-submersibles”, Conf on computer Aided. Design, Manuf and Oper in the Mar and Offshore Ind CADMO 1986, Washington. Santen JA (1999) “Jack-up model tests for dynamic effects on intact and damaged stability”, Seventh Int Conf The Jack-up Platform: Design, Constr and Oper, City Univ, London, Sept. Santen JA (2009) The use of energy build up to identify the most critical heeling axis direction for stability calculations for floating offshore structures, review of various methods. 12th Jack-Up Conf 2009 City Univ, London. Witz JA, Patel MH (July, 1985) “A pressure integration technique for hydrostatic analysis”, RINA suppl pap, Vol. 127. Couser P (2003) “A Software Developer’s Perspective of Stability Criteria”, 8th Int Conf on the Stab of Ships and Ocean Veh, STAB 2003, Madrid.
  • M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, 217 Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_11, © Springer Science+Business Media B.V. 2011 Some Remarks on Theoretical Modelling of Intact Stability N. Umeda*, Y. Ohkura*, S. Urano*, M. Hori* and H. Hashimoto* *Dept. of Naval Architecture and Ocean Engineering, Osaka University, 2-1, Yamadaoka, Suita, Osaka, 565-0871, JAPAN Abstract It is essential for realising risk-based design or performance-based criteria to establish theoretical modelling of extreme behaviours of an intact ship in waves up to capsizing. For this purpose, this paper summarises latest progresses in the researches conducted by the authors at Osaka University for improving theoretical modelling of intact stability in waves. Firstly, the methodology for calculating capsizing probability in beam wind and waves with piece-wise linear assumption is examined. Here a new formula covering both capsizing towards windward and leeward directions is provided. Secondly, for calculating restoring moment in longitudinal waves with sufficient accuracy, a hydrodynamic prediction method with wave-making and lifting components taken into account is proposed, and is reasonably well compared with captive model tests of a post Panamax container ship in head and following waves. This can be used in place of the Froude-Krylov prediction or captive tests for assessing the danger of parametric rolling for a practical purpose.Thirdly, for more efficiently identifying dangerous conditions in following and quartering waves, a numerical method for determining a heteroclinic bifurcation is proposed with its successful example to applied to surf- riding thresholds with nonlinear factors taken into account. 1 Introduction At the International Maritime Organisation (IMO), the review of Intact Stability Code started in 2002 and could open a door to use direct stability assessment with physical and theoretical modelling as an alternatives to prescriptive rules in the near future. Here quantitative accuracy of these modelling is essential. The benchmark testing of the 23rd International Towing Tank Conference (ITTC), however, revealed that existing theoretical models for predicting capsizing do have only qualitative accuracy. This means that the existing best models can predict whether a ship capsizes or not and how she does but they can not predict
  • N. Umeda et al.218 exactly when she does after the specified initial conditions. (Vassalos et al., 2002) Therefore, improvement of theoretical modelling is indispensable for its practical use. After the 23rd ITTC, the research team of Osaka University continues to improve its own theoretical models from various aspects. (Umeda, Hashimoto et al., 2003, Umeda et al., 2004, Hashimoto et al., 2004A, Hashimoto et al., 2004B) In this paper, some of their latest progress are presented to initiate the discussion at the stability workshop. These theoretical modelling to be investigated at Osaka University covers the dead ship condition, parametric rolling and broaching. The dead ship condition is related to the revision of the weather criterion and the others were the capsizing scenarios that the 23rd ITTC benchmark testing specified. 2 Capsizing Probability in Beam Wind and Waves Although a ship could capsize in following and quartering seas with much smaller wave steepness than in beam seas, it is still important to assess stability in beam seas. This is because dangerous situations in following and quartering seas can be avoided by ship operation but those in beam seas cannot be. For example, if all operational means such as propeller thrust and rudder control are lost, a ship that is almost longitudinally symmetric suffers bean wind and waves. Here, since a ship cannot escape from a severe sea state by herself, she should survive for sufficiently long duration. For guaranteeing survivability of such dead ship condition (IMO, 2001), the weather criterion was adopted by several administrations (Yamagata, 1959) and then by IMO (IMO, 2002). Since this criterion utilises some empirical methods to predict aero- and hydrodynamic coefficients, applicability of this criterion to new ship types such as a large passenger ship is rather questionable. However, because the probabilistic safety level of the weather criterion was adjusted by selecting the average wind velocity with casualty statistics, (Yamagata 1959) it is necessary to evaluate the safety level that the current weather criterion implicitly guarantees for conventional ships. (Umeda et al. 1992, Umeda & Yoshinari 2003) For this purpose, analytical methods to calculate capsizing probability in beam wind and waves are indispensable. This is because numerical or experimental methods require prohibitively many realisations to obtain the reliable value of capsizing probability for a practical ship, which should have very small capsizing probability. Among some existing analytical methods, a piece-wise linear approach proposed by Belenky (1993) seems to be the most promising because it utilises analytically-obtained exact but simple solutions of linear equations only. The authors investigate Belenky’s method by executing numerical calculation with his exact and simplified methods and extend his methods to the case in beam wind and waves with both capsizing in windward and leeward directions taken into account. As a result, some problems and their measures are found. Among
  • Some Remarks on Theoretical Modelling of Intact Stability 219 them this paper shows one problem that capsizing probability could exceed 1 when a sea state is extremely severe. According to Belenky (1993), the formula of capsizing probability for the duration T without beam wind is as follows:    moAl APTuTP   ;0)]exp(1[2 (1) where     11211 21 1      ppA (2) : apparent roll angle, m0: border of two roll angle regions, 1, 2: eigenvalues of the roll system in the second range, 1. 1 : initial values of roll angle and roll angular velocity in the second range, p1, 1p : initial values of forced roll angle and roll angular velocity in the second range, ul: expected number of up-crossing at mo for a unit time and PA: probability of positive value of the coefficient A when the up-crossing of  occurs. On the other hand, the authors propose the following formula for the case of wind and waves.       ]};0 ;0{exp[1 w TAPu APuTP moA moAl     (3) where uw is expected number of down-crossing at -mo for a unit time. If we ignore wind here, the following formula can be obtained:     ];02exp[1 TAPuTP moAl   (4) Therefore, P(T) defined here cannot be greater than 1. The authors applied the above new formula to a car carrier, whose principal particulars are shown in Table 1. Numerical results are presented in Fig. 1. Here the righting arm is approximated with two lines by keeping the metacentric height, the angle of vanishing stability and the dynamic stability from upright to the angle of vanishing stability because a time-varying external force induces capsizing in this scenario. The wind velocity, UT, is assumed to be constant and waves are done to be fully developed with this wind velocity and to be modelled with ITTC spectrum. PA are calculated with both exact and simplified methods. The former is derived from 3-dimensional Gaussian probabilistic density function; the latter is based on the assumption that no resonance occurs in the second region. The results demonstrate that capsizing probability tends to 1 when the sea state becomes
  • N. Umeda et al.220 extremely severe and Belenky’s simplified method slightly overestimates capsizing probability obtained by the exact method. Further discussion is available in the work of Kubo et al. (2010). Moreover, the effect of fluctuating wind has been incorporated for the capsizing probability calculation by the authors. (Francescutto, Umeda et al. 2004) Fig. 1 Capsizing probability of the car carrier in beam wind and waves by using the simplified and exact methods of the piece-wise linear approach. (Paroka Ohkura & Umeda 2006) Table 1. Principal dimensions of the car carrier Items Car carrier Length overall: Loa Length between perpendiculars: Lpp Breadth: B Draft: T Vertical centre of gravity: KG Metacentric height: GM Lateral projected area: AL Height to centre of lateral projected area: HC 190 m 180 m 32.20 m 8.925 m 14.105 m 1.300 m 4327.860 m2 11.827 m 15.00 1.00E+00 1.00E-04 1.00E-08 1.00E-12 1.00E-16 1.00E-20 1.00E-24 1.00E-28 1.00E-32 1.00E-36 25.00 35.00 45.00 55.00 U (m/s) Simplified Method CapsizingProbability Exact Method
  • Some Remarks on Theoretical Modelling of Intact Stability 221 3 Hydrodynamic Effect on the Change in GM Due to Waves At the 23rd ITTC benchmark testing even the most reliable mathematical model can provide only qualitative agreements with free-running model experiments for capsizing due to parametric rolling. As well known, parametric rolling is induced by the change in GM due to waves. Thus the accurate prediction of the change in GM due to waves is essential but all mathematical model in the benchmark testing predict the change in GM due to waves with the Froude-Krylov assumption. The captive model experiment for the ITTC A-1 Ship, as a follow-up of the ITTC benchmark testing, revealed that the Froude-Krylov calculation significantly overestimates the change in GM due to waves. (Umeda et al., 2004) Possible reason of the difference between the experiment and the Froude-Krylov calculation are hydrodynamic forces such as a wave-making effect and a hydrodynamic lift due to a heeled hull. For modelling such hydrodynamic effects on the change in GM due to waves, Boroday (1990) attempted to use a strip theory with the heel angle taken into account. He reported that good agreement with model experiment is obtained by ignoring wave-making damping components but the wave-making damping deteriorates the agreement. For finding a final conclusion on this issue, the authors reformulate a strip theory to calculate the change in GM due to waves for a heeled ship in longitudinal waves. Here all radiation components in the roll moment due to heave and pitch motions and the heel effect of the roll diffraction moment are consistently taken into account; the end terms are also included to explain hydrodynamic lift components. Two-dimensional hydrodynamic forces are estimated with an integral equation method with the Green function. The subject ship used here is a 6600TEU post- Panamax container ship, whose principal particulars are shown in Table 2. Its numerical results are shown together with the captive model test results in Fig. 2. While the Froude-Krylov calculation does not depend on the Froude number, the present calculation depends on the Froude number because of hydrodynamic effects. In following seas the Froude-Krylov calculation significantly overestimates the amplitude of GM variation but the present calculation well explains the reduction of the amplitude. In head seas the Froude-Krylov calculation overestimates the amplitude of the model runs with the Froude number of 0.1 but underestimates those with the Froude numbers of 0.2 and 0.3. The present calculation predicts the amplitude in the order obtained from the experiment. This improvement indicates that the hydrodynamic effect cannot be ignored when we accurately estimate the change in GM due to waves.
  • N. Umeda et al.222 Fig. 2 Amplitude of GM variation in following and head seas for the post Panamax container ship with the wave length to ship length ratio of 1.0. Here h/ and Fn indicates the wave steepness and the Froude number, respectively. And “cal. (FK)” means the Froude-Krylov calculation.
  • Some Remarks on Theoretical Modelling of Intact Stability 223 Table 2. Principal dimensions of the post Panamax container ship Items Containert Ship length : Lpp 283.8m breadth : B 42.8m depth : D 24.0m draught at FP : Tf 14.0m mean draught : T 14.0m draught at AP : Ta 14.0m block coefficient : Cb 0.630 pitch radius of gyration : yy/Lpp 0.239 longitudinal position of centre of gravity from the midship : xCG 5.74m aft metacentric height : GM 1.08m natural roll period : T 30.3 s. 4 Surf-Riding Threshold Broaching associated with surf-riding was also the capsizing scenario used in the 23rd ITTC benchmark testing for intact stability and quantitative disagreement with free-running model experiments was pointed out. In particular, the surf-riding threshold estimated with the existing model usually underestimates that from the experiment. Then some of the authors continue their effort to improve theoretical modelling by examining various nonlinear factors that had been ignored. As a result, they found that nonlinearity in the wave-induced surge force acting on a hull and the wave effect on propeller thrust are essential to improve the prediction accuracy of surf-riding threshold. (Hashimoto et al, 2004A) It is well established that the prediction of surf-riding threshold could depend on initial conditions if we simply apply numerical simulation in time domain but the surf-riding threshold for a self-propelled ship starting from sufficiently low propeller revolution does not depend on initial conditions. This is because onset of surf-riding or broaching can be regarded as a heteroclinic bifurcation. (Makov, 1969, Umeda, 1990, Spyrou, 1996, Umeda, 1999) That is, surf-riding or broaching occurs when an unstable invariant manifold of a saddle-type equilibrium in a phase space coincides with a stable invariant manifold of neighbouring saddle- type equilibrium. Although a perturbation method (Ananiev, 1966 ) and an exact method (Spyrou, 2001) are available for a simplest mathematical model, the intro- duction of nonlinear factors and coupling could prevent their application. A numerical technique to identify surf-riding threshold as a heteroclinic bifurcation can be applied to more complicated theoretical model. The technique reported so far, however, is not suitable for a computer algorithm. (Umeda, 1990, Umeda, 1999) Here it is necessary to calculate time series starting from a saddle-type equilibrium
  • N. Umeda et al.224 to other saddle-type one with the information from locally linearised systems. Thus, in this paper, a mathematical procedure suitable for a computer program is proposed with a successful example for following sea case and perspectives towards quartering sea case is added. For a simplicity sake, the uncoupled surge motion in following waves is investigated with the following mathematical model. );()(),;,()( aGWaGGx XuRnuTmm   (5) where G: horizontal displacement of the ship centre to a wave trough, m: ship mass, mx: ship added mass, u: ship velocity ( cuG  ), n: propeller revolution number, a: wave amplitude, T: propeller thrust, R: ship resistance and Xw: wave- induced surge force acting on a hull. The proposed procedure can be summarised as follows. First, by assuming a certain value of n, the equilibria from Equation (5) are identified and their eigenvalues and eigenvectors of the locally linearised systems at the equilibria are calculated. Here the equilibria correspond to surf-riding; one is stable and the other is unstable. Then we integrate Equation (5) with time from the point that is slightly shifted from an unstable equilibrium in the unstable direction of its eigenvector. And we also integrate Equation (5) with time backwards from the point slightly shifted from the neighbouring unstable equilibria in the stable direction of its eigenvector. Then we quantify the difference between two obtained values of u at a specified value of G as (n). Next, we apply the Newton method to find the parameter n* that (n*)  0. The obtained n* can be regarded as a heteroclinic bifurcation point. By using the relationship between the propeller revolution and ship speed in calm water, the nominal Froude number as the bifurcation point is determined. The above numerical method is used to determine surf-riding threshold for the ITTC A-2 ship with and without the nonlinear surge force and the wave effect on propeller thrust taken into account. Figs. 3-4 illustrate rapid convergence of the Newton method applied to their problem. The results shown in Fig. 5 show that the nonlinear surge force and the wave effect on propeller thrust significantly increase critical velocity for surf-riding. This means that these two nonlinear factors are essential to improve prediction accuracy for capsizing associated with surf-riding.
  • Some Remarks on Theoretical Modelling of Intact Stability 225 Fig. 3 Time domain simulation from saddle-type equilibria with time forwards and backwards with the wave steepness of 1/15 and the wave length to ship length ratio of 1.5. Here c and  are wave celerity and wave length, respectively: Fn indicates the nominal Froude number. Fig. 4 Convergence of the procedure for determining the surf-riding threshold. Here the wave steepness is 1/15 and the wave length to ship length ratio is 1.5. For predicting the broaching threshold, it is necessary to use an coupled surge- sway-yaw-roll-rudder motion. As a result, the stable invariant manifold becomes 7-dimensional while the unstable invarioant manifold is 1-dimensional. (Umeda, 1999) Thus the additional condition to be satisfied is that the vector normal to the stable invariant manifold should be orthogonal to the unstable invariant manifold. Numerical works based on this procedure were published later (Umeda et al., 2007)
  • N. Umeda et al.226 Fig. 5 Surf-riding threshold for the ITTC A-2 ship in following seas predicted by numerical procedure to identify a heteroclinic bifurcation with the wave steepness of 1/10. Here Fn and /L mean the nominal Froude number and the wave length to ship length ratio, respectively. The present method is one that takes nonlinearity in wave-induced surge force and the wave effect of propeller thrust into account and the conventional method ignores them. 5 Conclusions As follow-ups of the 23rd ITTC benchmark testing, the research team of Osaka University provides some improvements in theoretical modelling of intact stability from the viewpoints of probabilistic theory, hydrodynamics and nonlinear dynamics. The major outcomes are as follows: 1. The formula to calculate capsizing probability in a dead ship condition is presented as an extension of Belenky’s piece-wise linear approach. 2. Hydrodynamic modelling of the change in GM due to waves could improve agreements with captive model experiments. 3. A numerical algorithm to determine the surf-riding threshold is presented with some important nonlinear elements taken into account. The newly-obtained results provide much larger critical velocity for surf-riding than the conventional results. Acknowledgements This work was carried out as a research activity of the RR-S202 and RR-SP4 research panel of the Shipbuilding Research Association of Japan in the fiscal years of 2003 and 2004, funded by the Nippon Foundation. The work is partly supported by a Grant-in-Aid for Scientific Research of the Ministry of Education, Culture, Sports, 0.45 0.40 0.35 present method conventional method Fn l/L 0.30 0.25 0.8 1.0 1.2 1.4 1.6 1.8 2.0
  • Some Remarks on Theoretical Modelling of Intact Stability 227 Science and Technology of Japan (No. 15360465). The authors express their sincere gratitude to the above organizations. The authors also thank Dr. Gabriele Bulian for his effective advice on the formula of capsizing probability, which enabled us to correct Equations (3) and (4) in this version. References Ananiev DM (1966) “On Surf-Riding in Following Seas”, (in Russian) Transactions of Kryloc Soc, Belenky VL (1993) “A Capsizing Probability Computation Method” J of Ship Res, 37:3: 200-207. Boroday IK (1990) “Ship Stability in Waves: On the Problem of Righting Moment Estimations for Ships in Oblique Waves”, Proc of the 4th Int Conf Stab of Ships and Ocean Veh, Naples, II: 441-451. Francescutto A, N Umeda et al (2004) “Experiment-Supported Weather Criterion and Its Design Impact on Large Passenger Ships”, Proc of the 2nd Int Maritime Conf on Design for Safety, Hashimoto H, N Umeda and A Matsuda (2004A) “Importance of several nonlinear factors on broaching prediction”, J of Marine Sci and Technol, 9: 2: 80-93. Hashimoto H, N Umeda and A Matsuda (2004B) “Model Experiment on Heel-Induced Hydrodynamic Forces in Waves for Broaching Prediction”, Proc of the 7th Int Workshop on Stab and Oper Safety of Ships, Shanghai Jiao Tong Univ, pp. 144-155. IMO (2001) “SOLAS : consolidated edition, 2001”, 44. IMO (2002) “Code on Intact Stability for All Types of Ships Covered by IMO Instruments”. Waves Makov Y (1969) “Some Results of Theoretical Analysis of Surf-Riding in Following Seas” (in Russian) Transactions of Krylov Soc, 126: 124-128. Paroka D, Y Ohkura and N Umeda (2006) “Analytical Prediction of Capsizing Probability of a Ship in Beam Wind and Waves”, J of Ship Res, 50 187-195. Spyrou KJ (1996) “Dynamic Instability in Quartering Seas: the Behaviour of a Ship During Broaching” J of Ship Res, 40: 46-59. Spyrou KJ (2001) “Exact Analytical Solutions for Asymmetric Surging and Surf-Riding”, Proceedings of the 5th Int Workshop on Stab and Oper Safety of Ships, Univ of Trieste, 4:4:1-3. Umeda N (1990) “Probabilistic Study on Surf-riding of a Ship in Irregular Following Seas”, Proc of the 4th Int Conf on Stab of Ships and Ocean Veh, Naples, I: 336-343. Umeda N (1999) “Nonlinear Dynamics on Ship Capsizing due to Broaching in Following and Quartering Seas”, J of Marine Sci of Technol, 4: 16-26. Umeda N, H Hashimoto and A Matsuda (2003) “Broaching Prediction in the Light of an Enhanced Mathematical Model with Higher Order Terms Taken into Account”, J of Marine Sci and Technol, 7:145-155. Umeda N, H Hashimoto, D Vassalos, S Urano and K Okou (2004) “Nonlinear Dynamics on Parametric Roll Resonance with Realistic Numerical Modelling”, Int Shipbuilding Prog, 51: 2/3: 205-220. Umeda N, Y Ikeda and S Suzuki (1992) “Risk Analysis Applied to the Capsizing of High-Speed Craft in Beam Seas”, Proc of the 5th Int Symposium on the Practical Design of Ships and Mobile Units, Elsevier, 2:1131-1145. 13:169-176. Osaka Pref Univ, pp. 102-113. : Kubo T, E Maeda and N Umeda. (2010). “Theoretical Methodology for Quantifying Probability of Stability Failure for a Ship in Beam Wind and and its Numerical Validation”, Proceedings of 4th International Maritime Conference on Design for Safety, Fincantieri, pp. 1-8.
  • N. Umeda et al.228 Umeda N and K Yoshinari (2003) “Examination of IMO Weather Criterion in the Light of Capsizing Probability Calculation, Conf Proc of the Soc of Naval Archit of Japan, 1: 65-66. Vassalos D et al (2002) “The Specialist Committee on Prediction of Extreme Ship Motions and Capsizing –Final Report and Recommendations to the 23rd ITTC”, Proc of the 23rd ITTC, INSEAN (Rome) II: 611-649. Yamagata M (1959) “Standard of Stability Adopted in Archit, 101: 417-443. Umeda N, M Hori and H Hashimoto (2007), Theoretical Prediction of Broaching in the Light of Local and Global Bifurcation Analysis”, Int Shipbuilding Prog, 54:4, 269-281. “ Japan”, Transactions of the Inst of Naval
  • 3 Parametric Rolling
  • M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, 231 Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_12, © Springer Science+Business Media B.V. 2011 An Investigation on Head-Sea Parametric Rolling for Two Fishing Vessels Marcelo A.S. Neves*, Nelson A. Pérez**, Osvaldo M. Lorca***, Claudio A. * *LabOceano, COPPE/UFRJ, Brazil ** Abstract The paper describes an experimental and numerical investigation on the relevance of parametric resonance for two typical fishing vessels in head seas. The investigation is aimed at assessing the influence of the incorporation of a transom stern to the design. Results for different Froude numbers are discussed. The first region of resonance is investigated. Distinct metacentric heights and wave amplitudes are considered. Quite intense resonances are found to occur. These are associated with specific values of metacentric height, ship speed and stern shape. In order to analyse the experimental/numerical results, analytic consideration is given to distinct parameters affecting the dynamic process of roll amplification. The influences of heave, pitch, wave passage effect, speed and roll restoring characteristics are discussed. 1 Introduction Parametric resonance of ships in longitudinal regular waves is discussed in the paper, as this is an important particular situation. Results for two typical fishing vessels are compared. Head seas conditions are contemplated in the discussions. Recently, some papers have called attention to the problem of parametric excitation in head seas. (France et al. 2003) reported on strong roll amplification in the case of a container ship. (Dallinga et al. 1998) and (Luth and Dallinga 1999) reported on the development of head seas parametric resonance in cruise vessels. Previous investigations conducted by the authors, Pérez and Sanguinetti (1995), Neves, Pérez and Valerio (1999), indicated that in longitudinal regular waves, at zero speed of advance, resonant parametric amplification may result in large and dangerous rolling motions, particularly for fishing vessels with a transom stern in low metacentric height loading conditions. In the present paper, experimental results for different Froude numbers in head seas are discussed. This is a condition often encountered by fishing vessels, as while trying to maintain position in rough weather. Rodríguez Universidad Austral de Chile ***InterMoor Brazil