WEGEMT Short Course on:
"New Techniques for assessing and Quantifying Vessel Stability and Seakeeping Qualities"
Trondheim, 8-11 March 1993
" D Y N A M I C S T A B I L I T Y - P R O B A B I L I S T I C A P P R O A C H E S ' Alberto FRANCESGUTTO
Department of Naval Architecture, Qcean and Environmental Engineering University of Trieste
Via A. Valerio 10, 34127 TRIESTE (Italy)
ABSTRACT
The hydrodynamic part of ship safety is actually managed by means of prescriptive stability criteria based on static stability characteristics of the ship. The analysis reveals that this approach leads to many contradictions whose final effect is to some extent the delay of the formulation and adoption of higher safety standards. Only the development of a fully physical approach could allow a realistic evaluation of the probabilities of ship loss through the different mechanisms. The final goal will be an effectively unified approach to ship safety.
In the first part of the paper, the safety problem of fishing vessels is considered in connection with the possibility of their loss in waves. To this end, different dynamic mechanisms are studied in detail, ranging from capsizing to loss of control and/or broaching-to phenomena.
The stability of a family of fishing vessels is investigated in the light of the existing I.M.O. criteria and of the new method of energy-balance proposed by Strathclyde University is illustrated. Then the set of linear differential equations describing the dynamics of the antisymmetric motions of roll, yaw and sway is solved to look for possible unstable solutions. It results that, in general, I.M.O. statistical criterion is less restrictive than I.M.O. weather criterion and both fall into a region of approximately 20-30 percent of net area positive according to energy-balance method. An analysis of the stability of the antisymmetric motions reveals the possibility of dangerous phenomena depending on the trim condition.
For the same family, an analysis of the threshold for the onset of parametric rolling in a following sea reveals the extreme sensitivity of this type of vessels to such a particular capsizing mechanism.
These results are then critically reviewed in the second part of this paper to look for an answer to the currently unsatisfying state of art of small ship safety.
The loss of a ship is often a direct or indirect consequence of large amplitude rolling. In the third part of this paper nonlinear rolling is thus examined from a probabilistic point of view. The nonlinear rolling in a stochastic sea is analysed by means of an approximate perturbation method. The results indicate that the possibility of bifurcations to resonant or subharmonic large amplitude rolling is possible in a narrow band sea. A numerical time domain simulation confirms these results and gives an
indication on the probability denstities of states, that in the sinusoidal limit should recover the domains of attraction. The results can be of great interest in cargo mechanics modelling, an item relevant to ship safety.
INTRODUCTION
Safety at sea is a complex goal to achieve. This is due to the extreme variety of ships and structures and the different causes that can give rise to casualties at sea.
Depending on the dominant aspects and on the level of knowledge, these causes are divided in different categories. An important role is usually attributed to the structural aspects and to ship stability.
Stability is the part of Naval Architecture traditionally considered to give the tools for a correct approach to the safety aspects connected with the possibility of ship loss through the mechanisms of capsizing, sinking and surf ridinging/broaching-to.
As will be shown in the following, this approach and the safety rules based on it, are to some extent inconsistent. The introduction of a correct approach and consequently of the more appropriate name of "hydrodynamic part of ship safety", will improve the actually unsatisfying level of safety for life, cargo and environment at sea. In addition, this goal could probably be obtained without going further in the way of paying increased safety levels with unacceptable reductions of payload and/or operational capability. This could originate a positive feedback making more attractive the improvements of safety in ship design introducing the concept of "hydrodynamic safety performance" and a subsequent effective "design for safety".
This program rests on the development of a fully physical approach to ship safety through which the probabilities of the different dangerous phenomena can be computed. The concepts and methods of reliability could then be applied to obtain risk evaluation and control.
In this paper, the contradictions of current approach are highligted together with their negative effect on developments and progress in the field of ship safety.
Many aspects of ship hydrodynamics together with the consideration of ship structural aspects and their mutual interaction are involved in the previsions about ship safety. In particular, ship stability, manoeuvrability and their interactions, cargo mechanics, water on deck, loss of hull integrity, etc., can play an important role. Since it is difficult to deal with all these items in a short paper, so that, in the following, main consideration will be given to:
- IMO criteria with a typical application on a family of fishing vessels; - the method proposed at Strathclyde University regarding the energy balance in waves to take into account the loss of stability in waves;
- the method proposed at Brunei University regarding the stability of antisymmetric motions as one of the ways to take into account the possible interactions between transversal and directional stabilities;
- the nonlinear rolling in deterministic and stochastic sea to take into account the possible effects on cargo mechanics with shifting of cargo, loss of structural integrity, etc.
To have an idea of the importance of this discussion in the case of fishing vessels, a look to the statistics of casualties reported in Table 1 is sufficient. In Table 1 the casualties leading to capsizing and broaching (about 60% of casualties) regarding Japanese fishing vessels in the period 1973-77 [1] are analysed with respect to the causes.
1. IMO CRITERIA AND OTHER RECENTLY PROPOSED METHODS
2.1. Introduction
The problem of the safety of navigation at sea is very difficult to handle in particular for ships whose characteristic dimensions render them more sensitive to the action of the marine enviroment and, among these, a relevant place is resen/ed to fishing vessels [2,3]. The hull shape of a fishing boat varies greatly due to different local conditions, fishing methods, construction material, engine weights, distance to the fishing grounds and other factors. Thus, it is obviously very difficult to design a few standard hulls which are suitable for all conditions.
In the last decades, different organizations have collected and published results of theoretical calculations, model tests and full scale trial in an attempt to indicate the trend in the factors which influence stability, resistance, powering and seakeeping qualities of the vessels. The final goal is to reach conlusive indications on how to find the optimum hull shape when designing a new fishing boat. The statistical analysis of the data available is also intended for estimating the total performance of an existing design so it can be investigated if there is still room for any improvement.
At international level, IMO has devoted particular attention to small ships and in general to ships less than 100 m in length and, among these, a relevant place has been reserved to fishing vessels. Unfortunately, the stability rules adopted by the main Classification Societies still belong to a quite old approach, being mainly based on Rahola's statistical results requiring prescriptive characteristics of the curve of righting arm curve in calm water. The new proposals [4,5] suggest the extension of the weather criterion to such ships and are presently included in the Torremolinos Convention, which is yet not approved by a sufficient number of State Governments to become an international rule. It would be interesting to investigate the reason of this unsatisfactory state of art, even if economical and political choiches seem to constitute the major problem for further progress.
In the first part of this section, the stability of a family of the BSRA trawler series is investigated in the light of the existing criteria, i.e. the I.M.O. statistical criterion and the I.M.O. weather criterion. In the calculations, a group of 15 vessels was considered in order to determine the effect on stability of varying different design parameters.
In the second part of this section, the effect on stability of a longitudinal wave is taken into account following the Strathclyde approach and
considering the effect of the parametric resonance in following sea. Finally, the possibility of other ship loss mechanisms, connected with directional stability such as brbaching-td is examined through an analysis of the stability of antisymmetric motions.
1.2. The family of vessels
The numerical investigations were carried out on a family of fishing vessels known as the BSRA trawler series [6,7]. The parent form for the series, model XF, was chosen to represent a ship 150-ft. LBP x 26-ft. 4 in. moulded breadth x 13-ft. 2 in. moulded draught with a displacement of 847 tons salt water.
The hull forms of the series were derived from the parent form according to the following transformations:
- affine distortion of BfT (B/L, T/L) starting from ship XF for hull forms XG, WO, WP, 907, while keeping LW^^^ constant;
- affine distortion of UV^'^ (B/L, T/L) starting from ship XG for hull forms WS, WR, WO, while keeping B/T and V constant;
- variation of CB (CR) by conformal transformation of the sectional area curve, starting from ship XF for hull forms ZP, ZQ, but having main dimensions constant;
- same as above, starting from ship XF for hull forms 851, 852, while maintaining B/T and L/V'^ constant;
- variation of XCB ffom ship XG for hull forms 975, 977, 978, by modifying the sectional area curve only.
All the hulls of the family have been normalized to the same displacement through geometrical similarity and are fitted with similar superstructures independently on their length. Their main geometrical characteristics are given in Table 2. The forecastle has an extension of approximately 25% of the ship's length and the freeboard is sufficient to avoid deck immersion until a heeling angle of 12,5°.
1.3. I.M.O. Statistical criterion
The I.M.O. "statistical" criterion for fishing vessels of 24 meters in length and over has been endorsed at the "International Convention for Safety of Fishing Vessels" held at Torremolinos in 1977 [8,9]. The criterion, that derives from the pionieristic work of Rahola, is written in terms of stability standards based both on statistical and other analysis of casualty records and on the experience of different fishing fleets throughout the world. The Standards are expressed in terms of prescriptive values for certain key features of the righting arm curve and can be summarized as follows:
- Standard A: The initial metacentric height GM should be not less than 0.35 metres.
- Standard B: The area under the righting lever curve should be not less than 0.055 metrexradians up to 30°.
- Standard C: The area under the righting lever curve should be not less than 0.090 metrexradians up to 40° or up to an angle where the non-watertight openings come under water (whichever is less).
- Standard D: The area under the righting lever curve should be not less than 0.030 metrexradians between the angles of heel from 30° to 40° or such lesser angle mentioned under Standard C.
- Standard E: The maximum righting lever should occur at an angle of heel preferably exceeding 30° but not less than 25°.
- Standard F: The righting lever should be at least 0.20 metres at an angle of heel equal to or greather than 30°.
To aquire more information about the relative importance of the requirements introduced by the I.M.O. statistical criterion, we show in Table 3 [11] the maximum allowable height of the centre of gravity above the keel according to the different Standards. The results indicate the very different weight of the different Standards. In particular. Standard A, prescribing a minimum value for the metacentric heigth GM is by far the less resctrictive. Standard E, relative to the location of the maximum value of the curve GZ is generally the most restrictive. These results are in agreement with similar ones found relatively to the fishing vessels of the Ridgely-Nevitt [10].
In spite of its great simplicity, the statistical approach can be, and indeed is, criticized on the following basis:
- the statistics refers to a great variety of vessels, loading conditions and operation areas, so that the concept of "homogeneous specimen", i. e. the basis itself for the validity of a statistical approach can be questioned; otherwise, the specimen would belong to so specific situations to be hardly applicable in general;
- it constitutes an "a posteriori" (hindsight) approach, i. e. in the best hypothesis, it can only follow the trends indicated in the records of casualties at sea;
- no physical description of the effective interaction environment-ship is considered.
We will return on this point in Section 2.2. 1.4. I.M.O. Weather criterion
In the recommended I.M.O. "weather criterion" [8,9] for fishing vessels of 45 meters in length and over in unrestricted sen/ice, the ability of a ship to withstand the combined effects of beam wind and rolling should be demonstrated for each standard condition of loading, with reference to Fig.1, by means of the following procedure:
- The ship is subjected to a steady wind pressure acting perpendicular to the ship's centreline which results in a steady wind heeling lever (lwi).
- From the resultant angle of equilibrium (qo), the ship is assumed to roll owing to wave action to an angle of roll (qi) to windward. Attention should be paid to the effect of steady wind so that excessive resultant angles of heel are avoided. The angle of heel under action of steady wind (qo) should be limited to a certain angle to the satisfaction of the Administration. As a guide, 16° or 80% of the deck edge immersion, whichever is less is suggested.
- The ship is then subjected to a gust wind pressure which results in a gust wind heeling lever (IW2).
- Under these circumstances, area "B" should be equal to or greather than, area "A".
- Free surface effects should be accounted for in the standard conditions of loading.
Also in this case the stability of the vessels is investigated in the light of the proposed criterion. In order to simulate different loading conditions, in the calculations the height of the centre of gravity above the keel was varied until the stability requirement failed. As a result we obtain the maximum allowable height of the centre of gravity for each vessel according to the weather criterion.
In Table 4 [11], we show the maximum height of the centre of gravity above keel allowed by the statistical and the weather criterion respectively. It appears that the statistical criterion is less stringent than the weather criterion.
The weather criterium approach is preferred to the statistical one as it introduces a description of the interaction environment/ship, so that it appears more "physical" and not "a posteriori". This is only partly true, since the description is too much simplified to be sufficiently realistic, and limited to a particular mode of ship loss, i. e. the capsize mode. In fact, the results shown in Table 4 indicate that the two methods give comparable results, with a slightly lower maximum allowable center of gravity heights corresponding to the second. In the opinion of the authors this should upset definitely the statistical criterium.
1.5. The enerqv balance in waves
The method recently proposed by Strathclyde University [12] is based on the consideration of a time-dependent roll restoring moment. To this end, in addition to the combined effects of beam wind and rolling, also the effect of a following or quartering sea on the stability of the ship is taken into account. In the proposed procedure the righting arm is computed for different positions of a wave with respect to the vessel. The wave is assumed to have the same length of the ship and the encounter period equal to her natural rolling period. Then, an ultimate half roll is supposed to occur between a windward angle and an extreme leeward angle. For each position of the wave relative to the vessel the minimum righting arm curve during the ultimate half roll is obtained and the corresponding energy-balance, taking into account damping and wind, is computed to give the net area, as shown in Fig. 2. The percentage of time with net area positive during the passage of the wave along the ship is assumed as an indicator of the safety from capsizing. Futher details can be found in Ref.12.
In the application presented in this paper, the parameters describing the ultimate roll and the wind lever are esimated using the I.M.O. weather prescriptions. In the first application no roll damping was assumed to act. A rough estimate made on one of the vessels, indicated that the consideration of an equivalent linear damping realistic for these ships in the absence of antirolling devices, could increase the percentage of net area positive by about 20%.
The results relative to a group of hulls of the BSRA family to analyse the effect of the variation of the parameter B/T (the only that was found to be relevant) are shown in Fig. 3 [11], where the KGmax/D is plotted against B/T. As
one can see, both I.M.O. criteria lie in a region correspong to about 10-25% net area positive, whereas the request of 100% net area positive if much more restictive in terms of the maximum allowable height of the centre of gravity.
In Fig. 4, the KG/D is plotted against the Kempf's [13] nondimensional rolling period TK=TR(g/B)i/2 for the considered different hulls. Drawing the lines connecting the KGmax/D characteristic of the different hulls with respect to the different stability standards considered, one can see that the request of greater stability, for exemple a greater percentage of net area positive, corresponds to pushing the Kempf's period towards lower values.
Actually, it is not recommended to have TK<8 orTK>14. In the first case, the roll behaviour of the ship is too stiff, so that the oscillations have a moderate amplitude but higher frequency, i.e. great accelerations on the deck, rendering the operations difficult. On the contrary, a great value of TK gives rise to a cranck behaviour with large amplitudes and still a loss of operational capability (and of stability!). We will return on this point in Section 2.2.
1.6. The effect of parametric resonance in following sea
The effect of heave-roll coupling on the stability in waves was investigated by considering the so called parametric roll excitation. As known [14,15], theoretical predictions, confirmed by experiments, suggest that large amplitude rolling can be excited in the case of quartering and following seas when the encounter wave period is approximately half the rolling period and moreover a threshold condition correlating relative righting arm variation to roll damping is fulfilled. In design terms, the threshold condition expresses a correlation between sea intensity and both hull form geometry and weight distribution. By lowering the centre of gravity or introducing additional roll damping, one has the possibility to avoid the phenomenon.
Preliminary computations were done using a simplified, but common approach based on the assumption of linear time dependent rolling equation, so that the threshold is given in terms of the comparison of the relative GM variation in a longitudinal wave and the equivalent linear nondimensional damping. The isocarenic hypothesis was assumed, corresponding to consider the quasi-static approach. In the present case, the analysis revealed an extreme sensitivity of this class of vessels to parametric resonance. In fact, no realistic values for KG/D were found to prevent the phenomenon also in the case of vessels having bilge keels of area up to several percent of waterline area. Thus, the only way to avoid such dangerous situation is linked to the ability of the master, who must appropriately handle the ship at sea. A more detailed analysis taking into account the effective heave-roll coupling and the limiting effect on the roll amplitude played by the righting arm and damping nonlinearities, could change appreciably these conclusions. Moreover, being a "resonance" way to large amplitude rolling and eventually to capsizing, it is necessary, for its build-up, that a sufficient degree of autocorrelation be present in the wave train, so that it is tied to the appearance of a quite narrow band sea. The roll of a ship in irregular following sea is analysed by Dunwoody [16] that describes the effect of GM fluctuations as an equivalent roll damping reduction. The author discusses also the different stability limits with respect to the onset of parametric resonance connected with the different levels of stability of a stochastic dynamical system.
1.7. The stability of antisymmetric motions
Following a method developped at Brunei University [17,18], we consider a mathematical model suitable for the analysis of the stability of the antisymmetric motions of the ship that is, in order, sway, roll and yaw. This method could describe the possibility of ship losses due to broaching-to as it introduces a coupling between roll motion and directional stability. Since, we are interested in the onset of the deviations, the system of equations governing the antisymmetric motions may be written as:
m(v-HUr) = mg(t)+YvV+YvV-HYpp+Y(j)(l)+Yrr+Yrr Ixp-lxzf = KvV+KvV+Kpp+Kpp-pgVGM(t)+Krr+Krr Izr-lxzP = NvV-t-NvV-nNpp+Npp+Nrr+Nrr
The system is linear, the effect of the rudder is not considered; attention is focused mainly on the accelerated motions and the effect of trim variations is considered through the derivatives expressing yaw and sway dependence on yaw and sway velocity. Dimensionless parameters are used through a suitable transformation. In particular, the parameter GM/LFn2 js introduced to study the stabilizing effect of increasing the metacentric height and the destabilizing effect of increasing speed. No frequency dependence is considered, i.e. the slow motion derivatives approach is used.
The solution of a linear system of homogeneous equations undergoes a fast decay or gives rise to a non decaying oscillatory or diverging behaviour. Assumed a solution of the form
v(t)=yoe^^ r(t)=roe^^ (!)(t)=(l)oe'^^ p(t)=?^(t>{t) (})=|a+icü
for a non trivial solution to exist, the determinant of the system of algebraic equations obtained substituting the above expressions in the system of differential equations has to be zero. From this, the characteristic fourth degree equation is derived.
The solutions are the eigenvalues of the system, and generally two of them, M-i and ^2 are real and two M-S+ICDS are complex conjugate, corresponding to the fact that of the three motions only roll can have an oscillatory behaviour. Dynamic instability prevails when at least one of the real eigenvalues or the real part of the complex is positive, the last case corresponding to oscillatory dynamic instability.The second step is to analyse the eigenvectors trepresenting the amplitude of the motion solutions. Their moduli will be normalized to the yaw motion (put equal to unity) to have the possibility of individuating immediately which is the predominant. To each eigenvalue X, we have thus a set of three eigenvectors that represent respectively sway, roll and yaw amplitudes. If the eigenvalue is positive in that rangeof GM/LFn2, the predominant eigenvector indicates the motion that can give rise to dynamic instability.
In computing the stability of antisymmetric motions of the BSRA series of fishing vessels [ 1 9 ] , the hydrodynamic derivatives for yaw and sway terms depending on velocity were estimated from manoeuvrability [ 2 0 , 2 1 ] . Only very
rough estimates were possible for the terms depending on accelerations and those regarding roll motion. This makes absolutely necessary to consider the possibility of performing a thorough experimentation in this field.
In Fig. 5, the eigenvalues of the model XF for different values of the trim cofficient Y=T/Tm are reported. As one can see, the eigenvalues are very sensitive to trim variations. In particular, both and the real part ^3 of X,3 are positive in a wide range for Y<0, that is in the trim by bow conditions. No instability is detected in the normal trim by stern conditions A look at the eigenvectors reported in Fig.6 relatively to Y = - 0 . 2 , indicates that the system instability is yaw dominated in both senses (diverging and oscillatory) since the values of the a's and of the 5's (representing respectively normalized sway and roll eigenvectors) are less than unity (representing normalized yaw) for the first two eigenvalues, whereas the third indicates the possibility of a strong roll-dominated instability, particularly at low values of GM/LFn^ (low static stability or high speed).
In Fig.7 the same results of Fig.5 are reported for the model 9 0 7 . Here the range of dynamic instability is wider then for XF. This could appear strange if one considers that the model 9 0 7 is more stable than XF both following IMO criteria and Strathclyde method. The question is that stability and safety is a very complex problem and different approaches measure the resistance to different dangerous phenomena. Here, not the pure capsizing possibility is considered (of course only a fully nonlinear model could account for that phenomenon), but also loss of control and broaching to, although in the opinion of some authors the phenomena of directional instability and broaching-to are not so clearly connected [ 2 2 ] . Moreover, sometimes the instability appears connected with an inherently poor course-keeping ability [23]. The results are not given in terms of KGmax/D due to the uncertainty in the evaluation of the hydrodynamc coefficients.
2 . CRITICAL ANALYSIS OF CURRENT APPROACH TO THE HYDRODYNAMIC PART OF SHIP SAFEPi'
2 . 1 . Introduction
Existing rules for the assessment of safety of ships are contained in the "stability criteria" that in the following will be considered as known. A general critical analysis of these criteria, of the misleading use of the term "stability" and of the sometimes ambiguous role played by different actors interested in ship safety, is contained in Ref. [ 2 4 ] . Here the analysis is focussed on particular aspects of this approach that have negative influence on the development of new concepts and methods for the improvement of ship safety levels. From here on, except where differently stated, we will refer simply to ship safety instead of hydrodynamic part of ship safety.
It is customary to separate the case of intact ship from that of damaged ship. Both are based on a static approach. The most advanced formulation
regarding the damaged ship stability begin to include the description of the transient phase of flooding, where transient is still intended in a quasi-static approach.
2.2. The case of intact ship
The first serious approach, in the following indicated as IMO Statistical, originated from the work of Rahola published in 1939. At that time, the knowledge of the mechanims leading to capsize and broaching-to was poor and the availability of efficient tools for direct hydrodynamic calculations extremely limited. Hence a statistical approach that can be regarded as scientifically unsatisfactory but practically relevant.
The result is in the form of correlations expressing limiting values (prescriptions) for some characteristics of the righting arm of the ship with respect to fixed-trim, calm-sea, isocarenic, transversal inclinations.
Due to its relative simplicity and to its apparent relation to the physical mechanism of capsizing, the correlation was taken as a basis for a stability criterion that, with small adjustments (requiring years of discussions!) is still in use.
In the meantime, the need to reduce the number of casualties stimulated the research and the analysis of casualties. The result was the proposal of the adoption of a more "physical" criterion, i.e. the IMO Weather criterion. The more or less evident opposition of many bodies delayed the application of this rule and originated a complicate classification of ships in relation to the type, size and range of operation. In many cases, especially for the smaller ships (the most sensitive to the meteomarine action!) only very elementary calculations or no calculations at all is requested. The reasons of the delays and of the differentiation of the requests depending on ship size appa-rently based on the difficulty and length of involved computa-tions, really masks the attempt to avoid excessive reduction of paylod or operational capability.
This point worths a particular attention since it is usually assumed an obvious the fact that increasing safety means decreasing payload. As we will show, this connection is, at least partially an intrinsic consequence of wrong initial assumptions about safety.
The statistical criterion, in fact, is an "a posteriori" one. In addition, it expresses a correlation between observed ship losses, in a particular class of ships sailing a very restricted region, and computed static stability characte-ristics. The main criticisms can be sinthetically expressed as:
a) there is no direct connection between the requests and the needs. The criterion does not involve seakeeping qualities, the forcing effect of meteomarine environment, trim variations,coupling among different motions, course keeping ability, shifting of cargo, etc;
b) the statistical nature of the correlation averages between good and bad projects (with respect to safety), as a result, the good projects are penalized;
c) as long as the ship remains intact, the probability of casualty at sea is usually quite low as regards the mentioned mechanisms;
d) the request to be "simple".
Since ships are objects with a quite high index of reliability, this means that a few bad projects can be heavily conditioning. This means that even a small subset of casualties only explainable as a wrong correlation attempt as per a), can render the system almost "incompressible" in the sense that a very strong additional request (i.e. a very low KG reducing drastically the payload) contributes to a negligibly small increase in the observed safety at sea.
This could be advocated as a partial justification of the negative consideration paid by the owners to improving stability criteria.
To exemplify, the results of a parametric analysis conducted on a family of fishing vessels belonging to the BSRA series is discussed in some detail.
Firstly, the maximum allowable KG/D to satisfy both IMO criteria has been computed [11]. The ships of the chosen subset are all obtained from the first one through affine transformation. They have the same value of L / V / 3 while B/T increases from 1.97 to the extreme value 3.44 for the model 9 0 7 . Reported as a function of B/T, the results exhibit a regular trend for both IMO criteria. The value of KGmax/D increases with B/T. In Fig. 4 the same values of the heigth of the center of gravity are reported as a function of the Kempf's nondimensional rolling period TK=TRVg/B. This is an interesting parameter as regards ship operational capability, since it is not recommended to have T K < 8 because the roll behaviour is too stiff, so that the oscillations have moderate amplitude but high frequency corresponding to excessive deck accelerations. At the opposite extreme, a value T K > 1 4 corresponds to large amplitude rolling (and loss of stability in the traditional sense). Good operational capability corresponds to the condition 8 < T K < 1 4 . A look at Fig. 4 indicates clearly that the two IMO criteria give comparable results with an average value of T K - 8 . A more stringent request of stability pushes the ship toward a too stiff behaviour. For example, the use of the Strathclyde method corresponding to the energy balance taking into account the effect on transversal stability of a train of waves [ 1 2 ] , leads to T K = 6 if 1 0 0 % net area positive is requested.
As regards point c), it is important to note that often the loss of a ship is the result of a concatenation of causes, among which an important role is certainly played by the following scenarios:
- loop: large amplitude motions/accelerations shifting of cargo => stability degradation =>;
- large amplitude motions/accelerations => shifting of cargo => loss of structural integrity of the hull => damage;
- large amplitude motions/heavy weather => water on deck => stability degradation.
A ship with movable cargo (solid or liquid) or water on deck cannot be considered intact in the classical sense unless the proper dynamics of cargo and of the mutual ship/cargo interaction is taken into account. In the particular case of liquids with free surface, either in a container, as water on deck or
water in a compartment as a result of flooding after damage, the consideration of the classical correction to the different components of GM is contradictory because it introduces a mixed approach statical/pseudo-dynamical. The system, even if attention is focussed on rolling motion, is in this case a two degrees of freedom one. Its behaviour is also qualitatively different from that foreseable with the classical approach. As an example, the results of an experimental study of the behaviour of a fishing vessel with water on deck are reported in Ref. [25]. According to the classical theory the amount and location of water would contribute to a reduction of the metacentric heigth of about 50%. This does not account for the sharp change of natural rolling period that doubles. Moreover, the effect of water on deck is in this case stabilizing. But one does not neglect the fact that the effect of moving water is to some extent similar to that of passive antirolling tanks that change the location of the resonance peak giving generally origin to two peaks instead of one. It is not easy to conclude that the obsen/ed is a general trend without a careful examination of the possible variations of angle and frequency of encounter. The effect of sloshing of water in a compartment of a fishing vessel is actually understudy [26].
As regards point d), it has to be observed that the search to be simple implies a generally poor physical description of the mechanisms of ship loss. This is evident for the statistical approach, but a closer examination reveals that is a problem shared to a large extent by all approaches. As a result, an unknown percentage of the correlation between casualties and prescriptions remains unexplained, with the negative feedbacks already mentioned.
2.3. About "optimization"
The simultaneous analysis of the seakeeping and stability performances 4] on the same specimen of ships indicates a quite good degree of correlation between the two characteristics, as shown in Fig. 8. Another subset of ships belonging to the same family, composed of models XF, WS, WR, WQ, ZP, ZQ, 851, 852, 975, 977 and 978, once more obtained through affine transformation or conformal transformation of the sectional area curve has then been considered. In spite of being ships of the same type and comparable size and forms, the results are now completely different. A look at Fig. 9 indicates that the degree of correlation is very poor, if any.
It is not difficult to explain these drawbacks. The correlation is good when, for similar forms, the relevant parameter to be varied is B/T. This, in fact, is positively correlated with both the investigated characteristics: linear seakeeping (vertical motions) and static stability. Increasing B/T on the other hand, decreases the course keeping ability that has been called as a cause of broaching-to. In these conditions it is a very abstract thing to try to find an optimum project out of graphs like those of Figs. 8-9. The difficulties connected with rank optimization when trying to mix vertical and transversal ship motions and other characteristics, and thus the drawbacks to include stability in optimization, is also evident from analysis like that reported Fig. 9a, taken from Ref. [47].
3. STOCHASTIC ANALYSIS OF NONLINEAR ROLLING IN A NARROW BAND SEA
3.1. Introduction
All approaches to ship capsizing suffer in different measure of intrinsic limitations. Experimental studies can be quite realistic in the incorporation of the physical modelling and of the high nonlinearities, but are generally limited as regards the possibility of variation of significant parameters and consequently in the parametric inference. On the other hand, theoretical approaches allow for good analyses of parametric dependences tDut generally fall in a sort of "principle of indetermination", being less accurate in the forecasting of large amplitude motions as they apparently improve incorporating more details of the physical modelling. As a consequence, if one wants to describe large amplitude motions (stability), quite rough mathematical models have to be used, generally one or "one and a half" degrees of freedom (i.e. one fully nonlinear and some other quasi-linear). On the contrary, low
amplitude motions (seakeeping) can be approached by good analytico- • numerical schemes incorporating, linear or quasi-linear, hydrodynamic
characteristics of increasing complexity. These difficulties are slowly disappearing due to the advent of powerful techniques for the numerical handling of nonlinear and 3-dimensional fluid dynamics.
The practical need of relatively simple guidance rules for the design and operation of ships and ocean vehicles, generally known under the quite improper name of "stability criteria", formerly approached in static or quasi-static way, is now studied by means of nonlinear time domain simulations. These constitute a sort of "numerical experiment", whereas true experiments are now mainly used for validation porposes, except the cases where the physical modelling is so complicate that cannot be actually tackled efficiently by other means, as for exemple the study of the capsizing of vessels in very rough weather, including breaking waves, performed in the Norwegian Project "Stability and Safety for Vessels in Rough Weather".
It is never easy to design and interpret correctly an experiment, also in the case it is a numerical one. In particular, the recently discovered possibility of bifurcation scenarios, renders very complicate and possibly meaningless the results of time domain simulations. This paper is devoted to a statistical analysis of two of the bifurcation schemes of the nonlinear ship rolling in the stochastic domain.
Having mainly in mind the Stability problems, it is clear that a major role is played by the large amplitude rolling motion. Unfortunately, the hydro-mechanical modelling of ship loss is still incomplete, so that the connection between large amplitude rolling and capsizing is not very clear. In principle, the large amplitude rolling motion of a ship can, in fact, be a stable motion, provided it is included in some stability boundary. In practice, the ship is a very complicate system, so that many dramatic scenarios can appear once large amplitude rolling is in some way originated. Actually, it is not very simple to quantify the relative importance of these consequencies of large amplitude rolling. On the other hand, neither the probability of large amplitude rolling has been stated in a satisfactory way.
In previous papers [27-29], it has been shown that large amplitude rolling in non extreme seas can be a consequence of a jump between the antiresonant state and the resonant one, or between non resonant and subharmonic states, due to the strong deviations from linearity. This originates the possibily of bifurcations that can be effective as a consequence of some change in the parameters in the case of a deterministic excitation, and due to
the very nature of the oscillation when a narrow band stochastic excitation is considered.
Since the phenomenon is quite sudden, and the amplitude differences can be dramatic also in presence of non very intense excitation, a parametric research to find the probability of its occurrence was initiated. The goal is to increase the knowledge of the probability density function of the response in the following three cases of interest:
- bifurcation between the anti- and the resonant oscillation in the main resonance region in stochastic beam sea;
- bifurcation between non resonant and subharmonic in the first subharmonic region in stochastic beam sea;
- bifurcation between zero amplitude (or negligible one) and parametric suharmonic rolling in stochastic following or quartering sea.
As a consequence of the possibility of bifurcations, the probability density functions can be bimodal functions for some values of the parameters. This allows an evaluation of the probabilities of the considered different oscillation states, and in particular of the probability of the onset of large amplitude rolling.
In this paper, the attention will be devoted to the first two.
3.2 The sea description
In structural analysis a failure is usually regarded either as a direct consequence of an overloading or as a consequence of fatigue. This latter is considered as the effect of a periodic or in any case time-varying load of intensity lower that that implied in the first.
Capsizing and sinking are also failures, often improperly regarded as a pure hydrodynamic problem related to ship motions and stability, and thus of exclusive pertinence of Naval Architecture. In some limiting cases, and almost always in safety criteria, attention is mainly devoted to hydrostatic aspects only.
Actually, in the opinion of the author, the linking between the two aspects should be considered with greater attention, especially in the light of the fact that many cases of ship loss can be reconducted to structural failure induced by motions. This can be due to slamming, sloshing, cargo shifting. In particular this latter can be caused by overloading or fatigue in the lashings.
A part the slamming, that is connected with vertical motions, the other phenomena are related to rolling. This is the reason why large amplitude rolling motion is of interest not only when it determines by itself a stability overloading. It could be in principle a stable, although very unpleasant, oscillation regime, mainly affecting the ship's operational capability. Actually, the ship is not a rigid body, so that sloshing, shifting of cargo, a gust of wind [30] or water on deck [31] could be sufficient to worsen the stability conditions sufficiently to reach some non-return point.
Different scenarios can lead to large amplitude rolling. Among them, worth noting are the effect of extreme beam sea or the effect of a moderate
sea, beam or quartering, of appropriate frequency through one of the mechanisms described in the Introduction. Whereas the action of an extreme sea, with breaking waves, can actually be approached only by means of experimental approaches, in the following the attention will be devoted to the second case that we will refer as the 'resonance and bifurcation scenario'.
Resonance and bifurcations are a consequence of a certain degree of autocorrelation in the time record representing the stochastic excitation. This means, strictly speaking, that consistent parts of the record, or subrecords, have to exhibit pseudosinusoidal characteristics. Of course, we will not consider the extremely unrealistic case of pure sinusoidal or monochromatic sea.
High autocorrelation means small spectral bandwidth, so that attention will be focussed on the narrow band sea spectra, having in mind the fact that it can be sufficient that a few consecutive cycles exhibit similar characteristics to have resonance and possibly bifurcations. In a previous paper [32] it was shown that such spectra can be well represented by white noise filtered through cascades of linear filters (one or two can be appropriate). Limiting to the simpler case, one has the following representation of the spectrum:
Sf = YCOf2So/[(cOf2-Co2)2+co272;
obtained shaping Gaussian white noise of level So.(Fig. 10) The filter "damping" y represents the bandwidth and allows the excitation to recover the sinusoidal case as y-^O; cof is the centerpeak frequency. The excitation so represented has zero mean and variance Gf2=7tSo.
in the analytical part of this study, the narrow band sea will be represented in the Stratonovich's form, by means of a couple of slowly varying amplitude sinusoidal terms. The amplitudes will be represented by two independent Gaussian processes. Numerical computations will referte filtered white noise. In the very narrow band case, a square box filter was used, as it simplifies the choice of the cutting frequencies and reduces the number of harmonics required.
A direct analysis of the local autocorrelation was conducted on long time records used as excitation in the time domain simulations. The results indicated that for Y=0.005 the record is totally pseudosinusoidal within 10% accuracy, whereas for the peaked JONSWAP, corresponding to Y=0.14 the pseudosinusoidal aspect seems practically limited to short subrecords.
As regards the realizability of such narrow band spectra, we recall the long discussion leading to the formulation of the JONSWAP standard spectrum in the search of a more pronounced sharpness than in Pierson-Moskowitz, in agreement with many observations at sea. Now there is still some hint in this sense. Moreover, we have to remember the intrinsic accuracy limitation that is introduced in the formulation passing from observations to spectrum [33]. Finally, we have not to forget the fact that, when considering seakeeping or seakindliness, the analysis in terms of extreme and significant values is in general sufficient. The first relate to long time scales, whereas the second allow short time forecasts. When dealing with stability and safety from capsizing, on the other hand, we have to consider all the possible dangerous phenomena. These are not only connected with extreme values of the excitation, but also with pseudosinusoidality. This latter is a local (in time)
characteristic whose distribution in practical time records is to great extent unknown apart the very interesting approach to the description of the statistics of successive wave periods through a two-dimensional Weibull distnbution [34].
3.3. Analvtical results
Large amplitude, non extreme, rolling in beam sea can be computed through the use of analytical/numerical approaches based on relatively simplified models. In particular, the rolling motion can be considered uncoupled and described by a second order nonlinear equation of the type:
l ^ + D ( ( p , ^ ) + AGZ((p) = F(t)
that, dividing by I, representing the nonlinear quantities D and G through polynomials up to the third order and introducing as usual the natural
/ A ~ G M ^
frequency o)o='V —j—becomes [27]:
^ + (2^1+51 x2) ^ t 52 ^ coo^x + «3X3 . f(t)
The roll equation was written in the absolute rolling angle, since this is the only one meaningful in the stochastic case. It allows quite interesting computations in the frequency domain by means of the use of perturbation methods.
In particular, the use of the method of multiple scales allowed to obtain approximate expressions valid for the variances of the stationary stochastic state. Only the final results will be reported here. For more details on the analytical procedure, see Ref. [35-37].
In the region of synchronism cOf = (ÜQ-.
- t h e roll variance a as a function of the tuning ratio cof / COQ;
G,2= —j- [(co,2-(o,2 3 302)2+ (2c0o^ieq)2(1+y/2^eq)2]a
with an equivalent linear d a m p i n g ^igq given by:
^^eq = 1^ + ^ Seq o2 Sgq = (5i +3cOo252).
At the order of approximation implied in the used method C0f2-c0o2 = 2cOo(cof-o)o) and the above equation for a2 can be transformed in the following explicit expression for the frequency shift of the frequency response curve:
^ = t + J ^ + - ( ^ ( 1 + Y / 2 p i e q ) - {2G)oHeq)2{1+Y/2^eq)2)^^'
- the maximum value Om of the roll variance as a function of excitation variance Of can be obtained imposing the reality condition on the square root of the above expression:
(2c0o)2[(^ + ^ e q ^ m 2 ) 2 ( 1 +Y/2^eq)] =
A similar analysis conducted in the region of the first subharmonic cOf = SCOQ
gives the expressions reported in Ref. [41].
- the quantity GQ representing the roll variance in linear approximation and is valid for the frequency region far from resonances.
GQ2 =(1 +Y/2^) G,2 / [(a)o2-COf2)2 + (2^(0,)2(1 +Y/2^i)2] = a,2/(coo2-co,2)2
- the excitation amplitude threshold o^^^ for the onset of subharmonic oscillations as a function of tuning ratio is also given in Ref. [41].
Having in mind the analysis of rolling at intermediate to large amplitude (Xmax = 0.8 radians), the righting arm model was truncated to the cubic term. To get characteristic values of the nonlinear term coefficient (the natural frequency was set to the value COQ = 1, corresponding to an oscillation period of 271), a parametric research was conducted on a specimen of 1 3 ships in different loading conditions [29]. The results of a best fit of the cubic polynomial up to different heeling angles indicated that in the range of interest the values «3 = 4.0 and = -0.5 could be representative of different situations. The first is quite common in modern containerships, whereas the second is typical of most ships in loaded condition.
As regards the damping, intermediate values were chosen, except for the study of the subharmonic, where a low damping condition was, for the moment, assumed. A question a part could be that connected with the assumed model of damping moment. As a result of the perturbative analysis, an equivalent linear damping |j.eq 'S introduced. It depends on the actual roll variance G2, so that it is not convenient to use directly a linear model. More subtle is the distinction between the different nonlinear contributions that here too appear strictly mixed as in [38]. Both contributions were retained in the model used in the theoretical analysis in the light of the fact that there is still some experimental evidence of an angle dependence of damping; moreover, going to higher orders, the analysis can indicate a separate contribution.
The excitation intensity represented by Gf = 0 . 1 4 1 4 , corresponds to a sea of moderate intensity, well below the intensity corresponding to the breaking limit for the equivalent sinusoidal wave.
The deterministic expressions found in [27] are recovered in the limit Y->0 as can be checked by inspection. Due to the particuliarities of the employed method, using a representation based on two slowly varying amplitude sinusoidal functions and the multiple scales perturbation, it is not very easy to compare with other results valid for broader band excitation, such as the stochastic averaging.
The analytical results indicated that the bifurcation scheme is preserved also in the narrow band stochastic domain, as shown in Figs. 11-2.
An interesting question is constituted by the meaning of the different rolling variances, corresponding to the same intensity and tuning ratio, of Figs. 11 and Fig. 12, of the curve of maximum variances in synchronism
In the deterministic scheme, the solution was lead back to an initial values problem, so that the three solutions existing in some frequency range of the synchronism region can be divided in stable and unstable in the asymptotic sense. The Van der Pol plane representing the initial conditions can be divided in two regions containing the corresponding stable solution, called domains of attraction [28], separated by a separatrix curve passing through the unstable solution. In the absence of external perturbations changing some parameter, once started in one of the two domains, the solution will remain always in the same domain, leading to the stable solution as steady state. The curve of maximum amplitudes gives the highest of the two stable oscillation states. Similar considerations can be drawn for the oscillation in the region of the first subharmonic.
Giving up the deterministic scheme, the strict dependence on initial conditions has to be renounced to some extent. All the previous conclusions have to be reviewed in probabilistic terms, looking to the stochastic as a sort of continuously changing initial condition transient. On the other hand, one has also the distinction between transient and stationary stochastic. To clarify this complex matter, an extensive time domain numerical simulation is in progress. Only the preliminary results will be reported in the following.
3.4 Numerical results
The procedure consisted in the construction of a long record (tmax = 600 periods) of rolling motion through numerical integration of the differential equation of rolling motion. Then, this record was analysed in terms of filtered statistics of maxima (analysis of the envelope) to get the pseudosinusoidal probability density function (p.d.f.) [39-41].
The results for the synchronism region shown in Fig. 13 exhibit a good agreement with the frequency domain analytical values. In particular we can observe that the distribution is bimodal. This indicates that the two "stable" solutions are no more separated as in the deterministic case and the system jumps up and down between the two extreme states. On the other hand, the deterministic behaviour with domains of attraction has to be recovered in the sinusoidal limit ( y^O ). The effect of realistic different initial conditions on the p.d.f. can be seen in Fig. 14. It seems that the bimodality is preserved, but the relative importance of the two peaks changes. In the opinion of the author, this indicates that the sinusoidal limit is recovered through a displacement of the p.d. between the two peaks and a simultaneous sharpening of the same In the
limit Y^O this gives a p.d.f. consisting in one delta function located in correspondence to one of the two stable solutions.
The results in the region of the first subharmonic are shown in Fig. 15. The pdf is also in this case bimodal. The onset of the subharmonic is evidenced not only by the presence of the second bump in the pdf. The analysis of the time record in terms of the period of rolling (not reported in the figures) indicated wide intervals where the oscillation was pseudo-synchronous with the natural oscillations of the ship ( period = 27r/c0o ) that is characteristic of resonance, instead than the forcing period ( = 2p/Q)f = 2p/3o)o ).
The presented numerical results for the subharmonic region, for the moment, refer to a case with very narrow band and quite low roll damping ability. It seems that the onset of a subharmonic oscillation is restricted to such limiting cases. Anyway, it is difficult to reach a definite conclusion, so that the analysis is in progress.
3.5. The probabilitv densitv function
The probability density function for the response envelope is given in [39-40] for a Duffing oscillator (linear damping only) in the form:
P(G) = Const a exp{ 4cOf^ (1+Y/2^Leq)of^ 3
[ 2 ( — ^ ) 2 - H ( Y + 2 ^ e q ) 2 ] a d a }
For particular values of the parameters, this function can exhibit multiple extrema at the roots of dp/dG=0. In the limits of the different approximations implyed by the schemes adopted (cOf = COQ), these points coincide with the roots of previous equations. In particular, when this equation has three real solutions, the pdf has two maxima and a minimum, thus exhibiting a bimodal character. The probability distribution as shown in Fig. 16 shows a fundamentally different behaviour with respect to the Rayleigh distribution applicable to the linear and quasi-linear cases (Fig. 17). The multivaluedness disappears as one moves away from the multivaluedness frequency window on both sides. It also disappears when the excitation spectrum level or the nonlinear diameter are reduced. The number of real solutions is reduced to one only, this case being usually defined as a mild non-Gaussian behaviour [42-45].
The results can be of particular interest in the evaluation of fatigue behaviour of hull/hull integrity and cargo securing/shifting as indicated in Section 2.2 [46], an item of great interest in the consideration of ship safety, as indicated by Table 1.
REFERENCES
[ I ] Takaishi Y., "Consideration on the Dangerous Situations Leading to Capsize of Ships in Waves", Proceedings 2nd International Conference on Stability of Ships and Ocean Vehicles STAB'82, Tokyo, 1982, pp. 243-253. [2] Morrall, A., "Capsizing of Small Trawlers", Trans. RINA, Vol. 122, 1980, 71¬ 101.
[3] Duong, C , "Fishing Vessel Stability, a State of the Art Review", Bureau Veritas, Techn. Paper 88/2, 1988.
[4] Proceedings International Conference on the SAFESHIP Project, Ship Stability and Safety, RINA, 1986.
5] Nedrelid, T., Jullumstro, E., "The Nonwegian Research Project Stability and Safety for Vessels in Rough Weather", Proc. 3''d International Conference STAB'86, Gdansk, 1986, Vol.1, 145-155.
[6] Patullo, R.N.M., "The BSRA Trawler Series (Part I)", TRINA, Vol.107, 215, 1965.
[7] Patullo, R.N.M., "The BSRA Trawler Series (Part II)", TRINA, Vol.110, 151, 1968.
[8] International Conference on Safety of Fishing Vessels 1977, I.M.O., London 1983.
[9] Intact Stability Criteria for Passenger and Cargo Ships, I.M.O., London 1987.
[10] Boccadamo, G., Cassella, P., Mauro, S., Scamardella, A., "A New Methodology in Order to Verify the Ship's Stability in the Preliminary Design Stage", Proceedings International Symposium Prads'92, Newcastle upon Tyne, 1992, Vol. 2, pp. 1173-1186.
[ I I ] Francescutto, A., Nabergoj, R., "Analyse Parametrique de la Stabilité d'une Familie de Bateaux de Pêche", Bulletin de I'Association Technique Maritime et Aéronautique, Vol. 90, 1990, pp.63-81.
[12] Vassalos, D., "A Critical Look into the Development of Ship Stability Criteria Based on Work/Energy Balance", TRINA, Vol.128, 217, 1986.
[13] Norrby, R., "Stability Problems of Coastal Vessels", Int. Shipb. Progress, Vol. 11, 1964, pp. 121-132.
[14] Cardo, A., Francescutto, A., Nabergoj, R., "Subharmonic Oscillations in Nonlinear Rolling", Ocean Engng., Vol.11, 1984, pp.663-669.
[15] Allievi, A. G., galisal, S. M., Rohling, G. F., "Motions and Stability of a Fishing Vessel in Transverse and Longitudinal Seaways", SNAME Spring Meeting/STAR Symposium, New York, 1988, pp. 13-31.
[16] Dunwoody, A. B., "Roll of a Ship in Astern Seas - Response to GM Fluctuation", J. Ship Res., Vol. 33, 1989, 284-290.
[17] Bishop, R.E.D., Neves, M. de A.S., Price, W.G., "On the Dynamics of Ship Stability", Trans. RINA, 124, 1982, 285-302.
[18] Bishop, R.E.D., Price, W.G., Temarel, P., "On the Dangers of Trim by the Bow", TRINA, Vol. 131,1989, pp. 281 -303.
[19] Francescutto, A., Reggente, S., Armenio, V., "On the Possibility of Loss of Control and Broaching of Fishing Vessels", Symposium Technics and Technology in Fishing Vessels, Ancona, May 1989.
[20] Inoue, S., Hirano, M., Kijima, K., "Hydrodynamic Derivatives on Ship Maneouvrability", International Shipbuilding Progress, 28, 1981, 1-14.
[21] Fujino, M., "Lectures on Ship Manoeuvrability - Prediction of Manoeuvring Performance", University of Tokyo, Technical Report No. 6013, 1985.
[22] Motora, S., Fujino, M., Konoyagi, M., Ishida. S., Shimada, K., Maki, T., "A Consideration on the Mechanism of Occurrence of Broaching-to Phenomena", Naval Architecture and Ocean Engineering (Japan), 20, 1982, 92-107.
[23] Bao-an Y., "Ein Beitrag zur Beurteilung der Stabilitat schneller Schiffe bei gekoppelter Gier-, Quer- und Rollbewegung", Schiffstechnik, Vol. 31, 1984, pp. 22-42.
[24] Francescutto, A., "Is it Really Impossible to Design Safe Ships", Spring Meetings of The Royal Institution of Naval Architects, London, 27-29 April 1992, paper n. 3, To appear on Transactions of The Royal Institution of Naval Architects.
[25] Cardo, A., Francescutto, A., Zotti, I., Mattioli, R., "Experimental Study of the Effect of Water on Deck on the Stability of a Fishing Vessel", Proceedings International Symposium NAV'92, Geneva, July 1992, Vol. 1, pp. 3.3.1-3.3.14. [26] Cardo, A., Francescutto, A., Armenio, V., Contente, G., "Dynamic Effects of Liquids on Board on the Stability of a Fishing Vessel", In Preparation for OTRADNOYE'93, Kaliningrad, May 1993.
[27] Nabergoj, R., "Small Vessel Optimization for Increased Seakeeping and Stability Performance", Proceedings 4th International Conference on Stability of Ships and Ocean Vehicles - STAB'90, Napoli, 1990, Vol. 2, pp. 597-603. [27] Cardo, A., Francescutto, A., Nabergoj, R., "Ultraharmonics and Subharmonics in the Rolling Motion of a Ship: Steady-State Solution", International Shipbuilding Progress, Vol. 28, 1981, pp. 234-251.
[28] Cardo, A., Francescutto, A., Nabergoj, R., "Deterministic Nonlinear Rolling: A Critical Review", Bulletin de I'Association Téchnique Maritime ed Aéronautique, Vol. 85, 1985, pp. 119-141.
[29] Cardo, A., Francescutto, A., Nabergoj, R., "The Excitation Threshold and the Onset of Subharmonic Oscillations in Nonlinear Rolling", International
Shipbuilding Progress,Vol. 32, 1985, pp. 210-214.
[30] Dahle, E. Aa., Myrhaug, D., Dahl, S. J., "The effect of Wind on Small Vessels", Proc. International Conference STAB'90, Napoli, 1990, Vol. 1, pp. 191-199.
[31] Falzarano, J. M., Troesch, A. W., "Application of Modern Geometric Methods for Dynamical Systems to Problem of Vessel Capsizing with Water-on-deck". Proceedings 4th International Conference on Stability of Ships and Ocean Vehicles - STAB'90, Napoli, September 1990, pp. 565-572.
[32] Francescutto, A., Cardo, A., Contente, G., "On the Representation of Sea Spectra Through Linear Filters" (in Italian), Tecnica Italiana, Vol. 56, 1991, pp. 1-10.
[33] Guedes Scares, C , Trovao, M. F. S., "Influence of Wave Climate Modelling on the Long-term Prediction of Wave Induced Responses of Ship Structures", Proceedings of the lUTAM Symposium on Dynamics of Marine Vehicles and Structures in Waves, London 1990
[34] Myrhaug, D., Rue, H., "Note on a Joint Distribution of Successive Wave Periods", To appear.
[35] Rajan, S., Davies, H. G., "Multiple Time Scaling of the Response of a Duffing Oscillator to Narrow-Band Random Excitation", Journal of Sound and Vibration, Vol. 123, 1988, pp. 497-506.
[36] Davies, H., G., Rajan, S., "Random Superharmonic and Subharmonic Response: Multiple Time Scaling of a Duffing Oscillator", Journal of Sound and Vibration, Vol. 126, 1988, pp. 195-208.
[37] Francescutto, A., Nabergoj, R., "A Stochastic Analysis of Nonlinear Rolling in a Narrow Band Sea", Proceedings 18th ONR International Symposium on Naval Hydrodynamics, Ann Arbor, August 1990.
[38] Cardo, A., Francescutto, A., Nabergoj, R., "On Damping Models in Free and Forced Rolling Motion", Ocean Engineering, Vol. 9, 1982, pp. 171-179. [39] Davies, H. G., Liu, 0., "The Response Envelope Probability Density Function of a Duffing Oscillator with Narro Band Excitation", To appear on Journal of Sound and Vibration.
[40] Davies, H. G., Liu, Q., "On the Narrow Band Random Response pdf of a Nonlinear Oscillator", To appear.
[41] Francescutto, A., "On the Probability of Large Amplitude Rolling and Capsizing as a Consequence of Bifurcations", Proceedings 10th International Conference on Offshore Mechanics and Arctic Engineering 'OMAE', Stavanger, June 1991, Vol. 2, pp. 91-96.
[42] Ness, O. B., McHenry, G., Mathisen, J., Winterstein, S. R.(1989). "Nonlinear Analysis of Ship Rolling in Random Bean Waves", Proc. Research Workshop on Stochastic Mechanics, Technical University of Denmark, Technical Report N. 244/R, pp. 49-66.
[43] Juncher Jensen, J. (1989). "On Fatigue Damage due to Non-Gaussian Responses", Danish Center for Applied Mathematics and Mechanics, Report N. 389.
[44] Juncher Jensen, J. (1991). "Fatigue Analysis of Ship Hulls under Non-Gaussian Wave Loads", Marine Structures, Vol. 4, pp. 279-294.
[45] Francescutto, A., "Stochastic Modelling of Nonlinear Motions in the Presence of Narrow Band Excitation", Proceedings International Symposium on Offshore and Polar Engineering - ISOPE'92, San Francisco, June 1992, Vol. 3, pp. 554-558.
[46] Hutchison, B. L., "Cargo Mechanics (Application of Seakeeping-Revisited)", Marine Technoloqv, Vol. 23, 1986, pp. 230-241.
[47] Kishev, R., Dimitrova, S., Gaberova, M., "A Generalized Procedure for Rank Optimization Application to Ship Design", Proceedings International Symposium PRADS'89, Varna 1989, pp. 33.1-33.9.
Table 1. Main causes of loss for capsize and broaching regarding Japanese fishing vessels in the period 1973-77
Main Cause of Accident N
Overloading 9
Center of mass too high 13 Insufficient securing of cargo 1 Improper stowing of cargo 7
Cargo shifting 13
Openings not secured 9
Weack hatch covers 2
Hull damage 2
Water trapped on deck 29
Broaching 1
T a b l e 2. Main characteristics of the BSRA trawler s e r i e s . M O D E L B T D L / V 1 / 3 B / T C B C W X C B % BSRAIXF BSRAIXG BSRA I WO BSRA I WP BSRA I 907 BSRAI WS BSRA I WR BSRA I WQ BSRA II ZP BSRA II ZQ BSRA II 851 BSRA II 852 BSRA II 975 BSRA II 977 BSRA II 978 45.72 8.03 45.72 8.50 45.72 8.96 45.72 9.40 45.7210.62 41.01 8.98 43.36 8.73 48.09 8.30 45.72 8.03 45.72 8.03 45.72 8.32 45.72 7.82 45.72 8.50 45.72 8.50 45.72 8.50 4.07 3.85 3.64 3.48 3.09 4.07 3.95 3.75 4.08 4.06 4.23 3.96 3.69 3.72 3.74 4.95 4.77 4.63 4.51 4.21 5.04 4.90 4.46 4.97 4.95 5.15 4.81 4.77 4.76 4.75 4.85 4.85 4.85 4.85 4.85 4.35 4.60 5.10 4.92 4.76 4.85 4.85 4.85 4.85 4.85 1.97 2.21 2.46 2.71 3.44 2.22 2.22 2.22 1.97 1.97 1.96 1.97 2.30 2.29 2.28 0.562 0.561 0.562 0.562 0.560 0.559 0.560 0.561 0.521 0.593 0.521 0.593 0.584 0.580 0.577 0.774 0.776 0.776 0.777 0.777 0.772 0.765 0.775 0.742 0.806 0.743 0.805 0.816 0.795 0.790 0.969 0.960 0.962 0.965 0.949 0.678 1.095 0.959 0.888 0.906 0.892 0.903 4.689 0.693 -0.875 minimum maximum 41.01 7.82 48.0910.62 3.09 4.23 4.21 5.04 4.35 5.10 1.96 3.44 0.521 0.593 0.742 0.816 -0.875 4.689
Table 3a. KGmax (m) according to IMO statistical criterion. M O D E L A B S T A N D A R D C D E F BSRA I X F 3.488 3.315 3.233 3.172 3.216 3.166 BSRA 1 X G 3.588 3.423 3.320 3.232 3.209 3.238 BSRA 1 WO 3.769 3.588 3.455 3.327 3.227 3.357 BSRA 1 WP 3.984 3.775 3.614 3.449 3.269 3.505 BSRA 1 907 4.777 4.413 4.123 4.107 3.440 3.959 BSRA 1 WS 3.882 3.607 3.509 3.432 3.387 3.156 BSRA 1 WR 3.696 3.527 3.414 3.315 3.286 3.326 BSRA 1 WQ 3.490 3.333 3.238 3.165 3.134 3.438 BSRA II ZP 3.456 3.187 3.132 3.113 3.245 3.070 BSRA II ZQ 3.645 3.179 3.124 3.103 3.213 3.058 BSRA II 851 3.597 3.306 3.256 3.243 3.363 3.188 BSRA II 852 3.359 3.082 3.034 3.023 3.126 2.970 BSRA II 975 3.577 3.261 3.192 3.153 3.229 3.116 BSRA II 977 3.537 3.224 3.158 3.123 3.199 3.084 BSRA II 978 3.522 3.209 3.145 3.113 3.184 3.070
Table 3b. K G m a x / D according to IMO statistical criterion. M O D E L S T A N D A R D A B C D E F B S R A 1 X F 0.704 0.669 0.653 0.640 0.649 0.639 B S R A 1 X G 0.752 0.717 0.696 0.677 0.672 0.678 BSRA 1 WO 0.815 0.776 0.747 0.719 0.698 0.726 BSRA 1 WP 0.884 0.838 0.802 0.765 0.725 0.778 BSRA 1 907 1.134 1.048 0.979 0.975 0.817 0.940 BSRA 1 WS 0.770 0.715 0.696 0.681 0.672 0.626 BSRA1 WR 0.754 0.720 0.696 0.676 0.670 0.679 BSRA 1 WO 0.782 0.747 0.725 0.709 0.702 0.770 BSRA II ZP 0.696 0.641 0.630 0.626 0.653 0.617 BSRA II ZQ 0.700 0.642 0.631 0.627 0.649 0.618 BSRA II 851 0.698 0.642 0.632 0.629 0.653 0.619 BSRA II 852 0.698 0.640 0.630 0.628 0.677 0.617 BSRA II 975 0.749 0.683 0.669 0.661 0.677 0.653 BSRA II 977 0.743 0.677 0.663 0.656 0.672 0.648 BSRA II 978 0.742 0.676 0.662 0.655 0.670 0.646
Table 4. Maximum allowable height of the centre of gravity above keel. M O D E L IMO KGmax STATISTICAL KGmax/D IMO KGmax WEATHER KGmax/D BSRA IXF 3.172 0.640 3.005 0.606 BSRA IXG 3.209 0.672 3.046 0.638 BSRA 1 WO 3.227 0.698 3.107 0.672 BSRA 1 WP 3.269 0.725 3.225 0.716 BSRA 1 907 3.440 0.817 3.659 0.869 BSRA 1 WS 3.156 0.626 3.268 0.648 BSRA 1 WR 3.286 0.670 3.125 0.638 BSRA 1 WQ 3.134 0.702 2.968 0.665 BSRA II ZP 3.070 0.617 3.058 0.615 BSRA II ZQ 3.058 0.618 2.991 0.604 BSRA II 851 3.188 0.619 3.180 0.617 BSRA II 852 2.970 0.617 2.904 0.603 BSRA II 975 3.116 0.653 3.039 0.637 BSRA II 977 3.084 0.648 3.013 0.633 BSRA II 978 3.070 0.646 3.001 0.632
FIGURE CAPTIONS
Fig.1 - The I.M.O. weather criterion according to Ref.7. Fig. 2 - The method of energetic balance of Strathclyde.
Fig.3 - Comparison of K G m a x / D versus main design parameters for different
stability criteria.
Fig.4 - Comparison of K G m a x / D versus Kempf's adimensional rolling period
TR(g/B)i^ according to I.M.O. stability criteria.
Fig.5 - Variation of the eigenvalues of the characteristic equation for GM/LFn2=0.1 as a function of trim parameter y for model XF. Positive values of the eigenvalues indicate instability.
Fig.6 - Eigenvectors for the model XF in the trim by bow condition as a function of G M / L F n 2 . yaw is unitary, a indicates sway and 5 roll.
Fig.7 - Variation of the eigenvalues of the characteristic equation for GM/LFn2=0.1 as a function of trim parameter y for model 1907. Positive values of the eigenvalues indicate instability.
Fig. 8. Stability rank index relative to the application of energy balance method versus rank index for seakeeping (vertical motions) for the same family of fishing vessels represented in Fig.4.
Fig. 9. Stability rank index relative to the application of energy balance method versus rank index for seakeeping (vertical motions) for another family of fishing vessels of the BSRA series.
Fig. 10. Normalized JONSWAP spectrum corresponding to unitary value of the peack enhancement factor (solid curve). The dashed cun/e represents the fit obtained using the rational representation as a linear filter.
Fig. 11 Roll variance a as a function of tuning ratio cof/coo. The following values have been used for the parameters: «3=4.0, p.=0.05, af=0.1414 (0^=0.2). The number on the curves indicates the value of y.
Fig. 12 Roll variance G as a function of tuning ratio cof/coQ. The following values have been used for the parameters: a3=-0.5, ^=0.05, G{=0.1414 (0^=0.2). The number on the curves indicates the value of y.
Fig. 13. Comparison of the analytical results in the synchronism region with the time domain simulation. The following values have been used for the parameters: «3=4.0, ^=0.05, Gf=0.1414, y=0.02. The value Wf/coo = 1.55 was used in the numerical computations. Initial conditions corresponding to Xo=-0.3 and Xo=0.
Fig. 14. Dependence of the probability density function on the initial conditions in the synchronism region. Solid curve refers to Xo=0. and Xo=0., whereas dashed curve referes to xo=-0.3 and xo=0. The following values have been used for the parameters: «3=4.0, |i=0.05, Gf=0.1414, 7=0.005, cof/coo = 1.55. Fig. 15. probability density function in the region of first subharmonic. The following values have been used for the parameters: «3=4.0, |i=0.005, Gf=0.1414, 7=0.005, cOf/cüQ = 3.2. Initial conditions corresponding to Xo=0.15
and Xo=-0.15.
Fig. 16. Envelope probability density function in the presence of bifurcations as obtained from the analytical approach. The following values were used for the parameters: coo=1., }i=.05, Gf=.1414, 7=.02, cof/coo=1.40, «3=4.0.
Fig. 17. Envelope probability density function in absence of bifurcations as obtained from the analytical approach. The following values were used for the parameters: o}o=1., |i=.05, Gf=.1414, 7=.02, cof/coo=1.40, « 3 = 0 . (linear system).
