Homoclinic Phenomena in Ship Dynamics

Deterministic Occurrences and Their Stochastic Counterparts

The HOMSHIP research project is focused on a specific category of phenomena of extreme ship dynamics, manifested by motion instability accruing from the presence of strong system nonlinearity. Cases of extreme behaviour of ships are rare. Whenever they are encountered however they immediately capture public attention due to their potential for high consequences. Regulations quite often have an empirical basis and regulatory gaps inevitably arise from time for new or rare dangerous events. The HOMSHIP project is intended to generate original contributions concerning the nature of a certain class of ship motion instabilities that, despite being seemingly unrelated, have at their origin a homoclinic or heteroclinic bifurcation. It is hoped this effort to enhance the establishment of a modern point of view, for one of the most traditional fields of ship science.

Ships operating in a seaway often perform various unintended motions around their commanded courses. A control effort involving continuous adaptation of ship speed and heading is thus realized in practice, in order to diminish the effect of the random excitations incurred on the ship from her wind/wave environment. Ensuring that a ship can consistently reach her destination safely and efficiently is of paramount importance and several new regulations, rules and guidelines are developed every year, to direct design choices and improve operational practices. However, even for an idealized, parametrically described, stochastic seaway, solution of the above problem from first scientific principles is a formidable one.

The avoidance of dynamic instability in particular, has always been one of the fundamental concerns in ship design and operation. Instabilities can arise in more than one form. Most common is to consider the instability on the transversely (to the ship) vertical plane that is associated with the hazard of ship capsize. For example, instability might be incurred leading to cargo shift and/or capsize when encountering steep waves from the side ("beam-sea"); or in a longitudinal seaway, according to a variety of mechanisms such as "parametric rolling" and "pure-loss of stability" (Belenky & Sevastianov 2007). On the other hand, several operating ships are known to be prone to a different ("horizontal-plane") instability that could lead to a failure of course-keeping. The associated hazards are, in this case, collision and grounding. Combination of horizontal and transverse plane instabilities (in the sense that the first triggers the second) is also known to be possible during operation in a steep stern quartering sea, especially for ships of small to moderate size. Such an occurrence is termed as "broaching-to" (Spyrou 1996). Due to its very dynamic nature it has been considered as a constant peril of seafaring since the days of ships with sails (Spyrou 2010).

The relevance of a type of instability to a ship (reflected by the probability of occurrence) depends on a number of factors, such as, the ship hull specifics, the size and the loading condition. Also, her range of speeds may render a ship less or more prone to exhibiting a certain instability. At the design stage safety issues are tackled through compliance to prescriptive rules and regulations that essentially dictate the acceptable values for the ship parameters. The "performance-based" route is emerging as an alternative option of assessment, relying upon focused simulation and experiment (IMO 2001). Regulatory effort addresses however only the practical interface and, even that, in the short term, relying more on empirical data rather than on true understanding of processes. As a matter of fact, the field of ship dynamics still abounds with unsolved problems, notwithstanding that some could be relevant to ships that are already in operation.

Whilst the customary practice is to classify dynamic instabilities from a physical perspective (basically according to the associated hazard) in the context of the current research project an instability will be regarded as the reflection of a specific qualitative change that has taken place in the system’s state-space organization. A well-known class of generic nonlinear phenomena will be set in focus, with connecting feature that they all accrue from a certain typology of global bifurcation of dynamical systems. More specifically, the targeted instabilities will have in their core one of the known types of homoclinic or heteroclinic bifurcation, a phenomenon realised by the contact in state space of the stable and unstable manifolds of a single (or more) unstable limit sets (Guckenheimer & Holmes 1997). The significance of such phenomena for the extreme motions of sea-going ships has been supported by a number of recent works (an overview is found in the collection of papers edited by Spyrou & Thompson 2000). However the stochastic nature of the seaway has not been addressed up to now in considering these occurrences.

The current research project is intended to generate contributions in a number of areas that offspring from a common theme associated with extreme ship behaviour and phrased as "ship instabilities originating from a homoclinic/heteroclinic bifurcation". A number of semi-independent sub-projects will thus be defined, clustering around a main spine. Mathematical models, methods of analysis and even the general objectives will be particular to each individual sub-project, depending on background knowledge and the perceived priorities of research internationally. Three specific problems of extreme ship motions will be tackled, since these are known to us to be linked with such dynamical phenomena:

Beam – sea rolling under deterministic and stochastic excitation. We intend to consider a coupled system comprised of the rigid ship and the particles of granular material (grains, ore etc) carried in one or more holds. We are not aware of any previous research on this problem, at the intended level of modelling, despite the great importance of the topic. Special interest will be on the mechanics of cargo shift and the subsequent behaviour under near resonant excitation. It is known that the sustainable wave slope is lowered substantially even for a small amount of bias, generated by shift of cargo (Thompson 1997). It is also known that heteroclinic tangency phenomena severely undermine the safe domain of ship motions in state-space (Thompson & Soliman 1990). We are very interested at the interplay of the granular material flow with the phenomenon of heteroclinic tangency that is known to occur in ship rolling.

"Surf-riding behaviour" of a ship in a steep following/quartering stochastic seaway. It is already established that the phenomenon of surf-riding is the result of a homoclinic saddle connection (Spyrou 1996). Surf-riding is conducive to the loss of stability leading to capsize by the mechanism of "broaching-to". It has received quite a lot of attention internationally and it appears on the list of phenomena for which regulation is forthcoming (IMO 2011). However, whilst well understood in the deterministic case of harmonic wave excitation, very little could be said about its character under multi-frequency excitation, let alone under a fully stochastic wave environment. We intend to make the gradual transition towards consideration of more realistic wave forms.

Course keeping and turning ability of ships with large windage areas under extreme wind loading. We have unravelled two different cases of homoclinic saddle connection occurrence for this problem (Spyrou et al. 2007, Spyrou & Tigkas 2008). The first is the separation of turning from forward motion under a certain ("non-zero") setting of the rudder. This can be paralleled with the options for rotation or stationary behaviour that appear for a pendulum excited by a constant external torque. A second occurrence that we discovered concerns the destruction of parasitic oscillations on the horizontal plane. This could happen for a ship encountering strong wind from the bow and with the rudder angle fixed or weak rudder control. We want to know how these phenomena are manifested for more realistic (stochastic) wind loading.

The research has duration of 36 months (29 Sept. 2012 – 28 Sept. 2015) and it takes place at the School of Naval Architecture and Marine Engineering of the National Technical University. It is organized in four work packages (WPs). The three are specific to the above focus areas and they were intended to cluster around a core WP on the fundamental issues of homoclinic phenomena.

Principal investigator is Kostas J. Spyrou, NTUA Professor of Ship Dynamic Stability and Safety. The research team is comprised also of 1 post-doctoral scientist and 3 PhD students. 2 well-known international experts (S.R. Bishop, Professor of Nonlinear Dynamics, University College London; and Dr. V.L. Belenky, Principal Scientist at "David Taylor Model Basin", USA) have kindly accepted to have a consulting role in the project.

A newly acquired "shaking table" equipment capable of creating motions in 6 degrees of freedom and enabling deterministic and random vibrations of large amplitude will be used for performing granular material vibration experiments. Numerical codes will be developed, or expanded, in relation to all three identified focus areas.

Belenky, V., Weems, K.M., Bassler, C.C., Dipper, M.J., Campbell, B., Spyrou K., (2011) Approaches to rare events in stochastic dynamics of ships, *Probabilistic Engineering Mechanics*.

Corwin, E.I., Jaeger, H.M. & Nagel, S.R. (2005) Structural signature of jamming in granular media, *Nature* 435, 1075-1078.

Douady, S. & Fauve, S. (1988) Pattern selection in Faraday instability, *Europhysics Letters*, 6, 221-226.

Dhooge A., Govaerts W., Kuznetsov Yu.A., Mestrom W., Riet A.M. & Sautois B. (2003) MATCONT and CL_MATCONT: Continuation Toolboxes for MATLAB. Report of Gent and Utrecht Universities.

Feudel, U., Kuznetsov, S. & Pikovsky, A. (2003) *Strange Nonchaotic Attractors: Dynamics Between Order and Chaos in Quasiperiodically Forced Systems*, ISBN-10: 9812566333, World Scientific Series on Nonlinear Science, Singapore.

Guckenheimer, J. & Holmes, P. (1997) *Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields*, Corrected 5th printing, ISBN 0-387-90819-6, Springer-Verlag, New York.

IMO (2011) Report of the Working Group, Development of Second Generation Intact Stability Criteria, SLF 53, London.

Kan, M. (1990) Surging of large amplitude and surf-riding of ships in following seas, *in Naval Architecture and Ocean Engineering, The Society of Naval Architects of Japan*, 28, Ship and Ocean Foundation, Tokyo, 14 pages.

McDonald, S.W. Grebogi, C., Ott, E. & Yorke, J.A. (1985) Fractal basin boundaries, *Physica D*, 17, 125-153.

Poschel T. & Schwager T. (2005) *Computational Granular Dynamics*. ISBN: 978-3-540-21485-4, New York: Springer – Berlin, Heidelberg.

Simiu, E. (2002) *Chaotic Transitions in Deterministic and Stochastic Dynamical Systems*, ISBN 0-691-05094-5, Princeton University Press, New Jersey.

Spyrou, K. J. (1996) Dynamic instability in quartering seas: The behaviour of a ship during broaching, *Journal of Ship Research*, SNAME, 40, 1, 46-59.

Spyrou, K.J. & Thompson, J.M.T. (2000) The nonlinear dynamics of ship motions: a field overview and some recent developments. *Philosophical transactions of the Royal Society, Series A*, London, 358 ,1735-1760.

Spyrou, K.J. & Tigkas, I. (2008) Nonlinear dynamics of ship steering behaviour under environmental excitations. Selected article (with further peer review) from: *IUTAM Symposium on Fluid-Structure Interaction in Ocean Engineering*, Springer (ed. E. Kreuzer), Berlin, 261-272, ISBN 978-1-4020-8629-8.

Spyrou, K.J. & Tigkas, I. (2011) Nonlinear surge dynamics of a ship in astern seas: "Continuation analysis" of periodic states with hydrodynamic memory, *Journal of Ship Research*, 55, 1, 19-28.

Spyrou, K.J., Tigkas, I. & Hatzis, A. (2007) Dynamics of a ship steering in wind revisited, *Journal of Ship Research*, 51, 160-173.

Themelis, N. & Spyrou, K. (2007) Probabilistic assessment of ship stability, *Transactions, Society of Naval Architects and Marine Engineers*, (SNAME), 115, 181-206.

Thompson, J.M.T. Rainey, R.C.T. & Soliman M.S. (1992) Mechanics of ship capsize under direct and parametric wave excitation, *Philosophical Transactions A*, The Royal Society, 338, 471-490.

Troesch, A.W. & Falzarano, J. M. (1993) Modern nonlinear dynamical analysis of vertical plane roll motion of planning hulls, *Journal of Ship Research*, 37, 189-199.

Wiggins, S. (1988) *Global Bifurcations and Chaos - Analytical Methods*, ISBN-10: 3540967753, Springer-Verlag, Berlin.

**Kostas Spyrou** is Professor of Ship Dynamic Stability and Safety at NTUA's School of Naval Architecture and Marine Engineering where he has also the position of Dean. He has worked in the field of nonlinear ship dynamics for 25 years.

**Stephen Bishop** is Professor of Nonlinear Dynamics at University College London (UCL). He is an international expert on nonlinear systems analysis, for many years Manager of UCL’s Centre for Nonlinear Dynamics and Its Applications.

**Vadim Belenky** is Principal Scientist at “David Taylor Model Basin”, USA. He is an international expert on probabilistic methods and their application to ship dynamics.

**Nikos Themelis** is a Naval Architect and Marine Engineer graduated (2002) from NTUA, where he also wrote his PhD thesis (2008) on a probabilistic assessment of ship dynamic stability. He has worked on several EU and national projects related with ship dynamics and safety.

**Christos Spandonidis** is an Electronic Engineer graduate of the Hellenic Air Force Academy. He currently studies towards the PhD degree on the behaviour of granular materials under realistic ship-like excitations applying mathematical modelling as well as experimental techniques.

**Panagiotis Anastopoulos** is a Naval Architect and Marine Engineer who graduated from NTUA in 2012. He studies towards a PhD in the assessment of ship stability under stochastic wave excitation modelled by wave groups theory.

**Ioannis Kontolefas** is a Naval Architect and Marine Engineer who graduated from NTUA in 2013. He currently studies towards the PhD degree in the field of nonlinear dynamics of surf-riding and manoeuvring under multichromatic wave excitation.

Four technical reports have been produced in the HOMSHIP project (relating to Tasks 2, 3, 4 and 5) and they are available upon request. Dissemination activities, with a listing of papers produced so far, are described below in the report of Task 1.

Objective: To improve fundamental understanding about homoclinic/heteroclinic phenomena associated with ship dynamic behaviour.

Objective: Theoretical and experimental investigation of cargo shift and its effect on capsizability on the basis of a coupled ship/granular material system under realistic excitations.

Objective: Adaptation/reformulation, of the surf-riding theory in probabilistic context.

Objective: To consider how the two identified homoclinic phenomena occurring under steady wind loading could appear for a more realistically modelled wind