Peculiarities of Motion of Ship with Low Buoyancy on Asymmetrical Random Waves

Transcription

1 Peculiarities of Motion of Ship with Low Buoyancy on Asymmetrical Random Waves Degtyarev A.B., Boukhanovsky A.V. Institute for High Performance Computing and Data Bases, 118 Fontanka, 1985 St.Petersburg, Russia ABSTRACT In the report rolling and heaving of ship in beam sea are considered. Method of ship motion investigation is imitative modelling. We use autoregressive method with correction of correlation function via non-gaussian character of asymmetrical random waves for its modelling. For ship motion modelling different model equations of rolling and heaving are used. Different situations are considered. The most dangerous (from stability point of view) of them are chosen. Calculations are compared with intact ship motion and with motion on symmetrical waves. 1. INTRODUCTION As it was noted at the 6th STAB conference in Varna (Bulgaria) universal nonlinear approach traditionally used for investigation of rolling is now being applied in investigation of other seaworthiness problems. It is related with development of ship and ocean vehicles of new unconventional architecture. There are many well-known papers devoted to different problems in nonlinear rolling. Growing interest to nonlinearity in ship rolling is related with possible large inclinations when qualitatively new peculiarities could appear. It was V.V.Lugovsky [1] who carefully investigated nonlinear rolling. The authors of this article have discussed application of different methods for ship rolling problem in random sea in []. Besides papers mentioned in this work it is necessary to note latest investigations of K.J.Spyrou [3, 4 and others], J.M.T.Thompson [5], M.Kan [6], K.Tanizawa [7], A. Francescutto [8], etc. All these works are devoted to study of different new qualitative rolling scenarios that could not be analysed by use of methods of linear or quasilinear systems. At the same time we

2 cannot consider only one kind of ship motion as nonlinear. Heaving, pitching, swaying, etc. in this case also have nonlinear character. We can only say that these nonlinear phenomena are weaker than rolling nonlinearity. However, it is well known that influence of other kinds of ship oscillations on rolling could follow dangerous motion regimes. So in couple motion (heaving and rolling) unstable oscillations could appear. Such phenomenon is parametric resonance. It can follow ship capsizing. A parametric oscillation is a very subtle phenomenon. It depends on many factors (so it is rare phenomenon) but has grave consequences. In previous papers [ and other] the authors have considered parametric resonance of convenient ship in random waves in different conditions. This paper is devoted to heaving consideration of inconvenient ship. It is underwater vehicle (UV) at the surface or damaged convenient ship with low buoyancy. Nonlinear draft dependence of underwater volume due to cylinder-like form (to distinguish from traditional ship types) is conditioned nonlinear vertical motion.. MATHEMATICAL MODEL OF UV WITH LOW BUOYANCY IN SEA WAVES Hull form of modern UV is rather complicated (fin stabilisers, tunnels, escape hatches and hatch houses). So let us investigate quality behaviour of the object related with nonlinear variation of waterplane area with draft increasing with the help of simplified physical model. We consider oscillation of round cylinder with radius R, length L and z-coordinate of centre of gravity Z g on the surface of inviscid boundless fluid. Let us also consider only two-dimensional beam sea. In this case we can use only equations of heaving and rolling for solution. We can exclude swaying from consideration because it is not qualitative significant. In this case taking swaying into account could correct qualitative result but do not bring qualitative leap. Co-ordinate system is shown in figure Forces and moments acting on UV on calm water Characteristics of cylinder position (see fig.1) are determined by relationship of draft T and waterline breadth B in considered time moment. Difference of maximum breadth (R) from current one B (i.e. degree of problem nonlinearity) is characterised by parameter Z p =T-R. In accordance with [9,1] the following forces and moments act on oscillated plate contour: Force and moment of inertia D Fi = + λ Mi [ Ixx ] g, = 33 Θ + λ44 (1) where D is UV displacement; I xx is moment of inertia with respect to axis X, is heave ordinate; Θ is roll angle (see fig.1). Exact formulas of added masses for cylinder for heaving λ 33 and rolling λ 44 are obtained in [9]. Force and moment of damping F d = µ, M = µ Θ () 33 d 44

3 Approximate formulas for damping coefficients µ 33 and µ 44 are given in [9]. Linearized form () takes into account mainly energy dissipation due to pressure redistribution on UV hull and does not take into account damping due to vortex emergence. Nevertheless such form for damping components is more acceptable, because such approach allows to consider nonlinear effects stipulated by peculiarities only restoring and exciting forces and moments. Restoring force and moment In hydrostatic, restoring force appearing in shift of co-ordinate centre O 1 on value Z p + relative O is equal to R Fr = γv( ) = γl + Z arcsin R R p ( ) ( ) Zp arcsin + R p p p p + Z R Z + Z R Z, (3) or using series near =Z p and taking into account the fact that restoring moment is constant when >R, we have F r γπr L, > R ( ) = Zp 8R γlb + O 4 B 3B ( ) 3 4, R where γ is specific gravity of water. Restoring moment of rolling for rounded cylinder is not dependent on. Then we can present it as [11]: M r (Θ)=D(R-Z g ) sin Θ (5) and correspondent series expansion is where h is initial metacentric height... Exciting force and moment M ( Θ) r Dh Θ Θ 3 Θ 5 7 = + + O( Θ ) 6 1 (4). (6) Let us consider Froude-Krylov approach for calculation of exciting component. For regular wave w () t = cos( ω t+ ε), (7) we can obtain the following expression for exciting vertical force R Fw = γκ V( w) = γl or using expansion (4), we have arcsin + Z R R ( ) ( ) Z arcsin + R w p p w + Zp R Zp + w Zp R Zp (8)

4 F r ( ) = κ w γπr L, > R Zp 8R γlb w + O 4 B 3B ( ) 3 4 w w w, R (9) where κ is reduction coefficient for heaving. In the first approximation we can present it as ratio of pressure forces acting on UV hull stipulated by orbital fluid particle movement Q and amplitude of exciting force Q : ( p ξ) R t [ kz ( )] [ kt] dt [ K p = + ] 1 Q κ ( k, ) exp ξ exp Q B R t B K o = = + K K 1 = = Z + ξ p Z + ξ + R p Z + ξ p z Z ξ R z Z z Z ξ ( p ξ) R z Z o, o < ( R Zp) ξ =, o ( R Zp) p p exp [ ] exp Z ξ p kz dz [ ] kz dz Term K 1 corresponds to wave pressure on UV hull with negative resulting force, and K is corresponds to positive one. Value ξ gives elevation of wave crest ( ) over waterline to the extent of hull diameter (this value influences wave pressure distribution and buoyancy force), h is elevation of wave crest over hull (buoyancy force is not changed yet, only wave pressure is recombined). Assessment of κ could be corrected if we determine value ξ as root y * of transcendental equation (1) r cos ky = R y Z ; ξ = R y Z (11) o p p In comparison with classical presentation of heaving reduction coefficients for conventional ships [11,1] value (1) has essentially other properties first of all due to low buoyancy of UV: reduction coefficient depends not only on wave frequency but on amplitude as well; beginning from certain value of wave number k, K dominates on K 1, i.e. reduction coefficient changes its sign (consequently exciting force and wave (7) are in opposition). The wave number dependence on the reduction coefficients (1)-(11) is given in fig. for different waterlines. One can see that reduction coefficient for small draft of cylinder is similar to coefficient for conventional ships, but coefficient changes sign when draft is deep. Exciting rolling moment in first approximation is presented in form [1,13]: M = ( I Α ) α (1) w Here α is wave slope (α d w /dx). Value A is determined by integration of underwater volume, it has reduction coefficient functions: xx

5 ( 5 p ) A y dy L R 4 R 4 Zp Zp = ρ = ρ π + arcsin + R Zp R Z (13) 8 4 R 1 V We emphasise that exciting moment due to heaving is of special importance for nonlinear rolling. This additional moment could initiates parametric oscillation. In accordance with [13] and (5)-(6) parametric term in relative co-ordinates has the following form: Mp( ϑ,, w) Dh ϑ ϑ 3 ϑ 5 = + w + γzpv( ) ϑ 6 1 where ϑ = Θ α is relative roll angle..3. Sea waves UV can find its way into different wave and weather conditions: wind waves, swell, complex sea, heavy sea or steep wave. Unfavourable conditions could have place during submerging or emerging. Due to variability of wind and variance properties of environment form of individual waves is continuously changed. So waves following one after another are not identical, and there are no exact periodicities of waves described by model (7). Longuet-Higgins model is traditionally applied for reproduction of random waves as realisation of gaussian stochastic process t [14]. However wave models of autoregressive-moving average [15-18] are more suitable for sea with complicated spectral structure (complex sea). Such models have better stability and convergence characteristics. (14) N = Φ + Θ ε t i t i j t j i = 1 j = P (15) Here Φ i are autoregressive parameters, Θ j are moving average parameters, ε is white noise with boundless divisible distribution law. Autoregressive model (P=) is in most common use for such phenomenon. In this case coefficients Φ j are uniquely determined by correlation function of initial process with the help of Yule-Walker equations system: N K ( i ) = Φ K ( i j ); i = 13,,,... (16) j = 1 j Variance of white noise could be determined from zero equation of system (16) or from spectral relation (if we take into account stochastic dispersion of K (j ) assessments in (16)): σ π K dω = ; J = ; Φ J N Φ exp ε π k = j [ ijω ] Here σ is white noise variance, is time series quantization. 1 (17)

6 Integral in denominator of formula (17) is numerically calculated after autoregressive parameters estimation. Waves take asymmetrical form when wave making conditions intensively change (strong choppy wind for example). In this case hollows become more plane, and crests become more acute. Thus distribution F (z) of wave ordinate (t) now is nongaussian. It requires to apply nonlinear transformation z=f(y) to model process y(t) of type (15). This transformation corresponds to solution of transcendent equation F = Φ(y) (18) where Φ(y) is univariate integral function of gaussian distribution, or the following R expression for mesh function { y k }, { z } = k k F ( zk ) = t exp 1 dt π. (19) However, correlation function is changed in any nonlinear transformation of stochastic process. In order to avoid it preliminary transformation of correlation function with the help of Edjgeworth s series expansion is used. Covariance function K (τ) (or spectral density S (ω)) of stationary stochastic process z with distribution law similar gaussian could be presented as Gram-Charlier serie expansion on powers of covariance function K y (τ) (spectral density S y (ω)) of gaussian process y: K C K m y ( τ) z( τ) = m, m= m! () S C S m y ( ω) z( ω) = m, m= m! (1) where 1 y Cm = f( y) Hm( y)exp( ) dy, π () and H m (x) are Hermit polynomials. Values of model correlation function K y (τ) (spectral density S y (ω)) are obtained with the help of known values of coefficients C m and values of covariance function for each shift value τ (or frequency ω for spectral density) as solution of nonlinear algebraic equation: m R : k= m Ck k K k K Ck k S k S y ( ) z( ), y z! τ τ = ( ) ( )! ω ω = k = k = y k (3) Expressions (3) are polynomials of power m relative K y (τ) and S y (ω). Solution of the problem is real root of polynomial in interval [-D, D ]. Thus model (3) compares model gaussian process with redistributed energy on frequencies to initial nongaussian process z. Redistribution of energy in accordance with (3) offers to model spectrum S (ω) that identical to initial spectrum S (ω) after nonlinear transformation. Realisations of gaussian and nongaussian waves in fixed point reproduced with the help of described model are shown in fig.3.

7 Continuous records of sea waves obtained in near Gelendgik (Black Sea) were used for identification and testing of proposed model. Measurements were carried out in the frames of international project TU WAVES/NATO. Results are shown in [19]..4. Equation system for UV rolling and heaving Let us consider the following system for UV rolling and heaving proposed by V.Lugovsky [13] and based on the above reasons. D g + λ + µ γv( ) = γκ( ω) V( w ) ω g + γ ϑ = α ( I + λ ) ϑ + µ ϑ+ Dl( ϑ) 1 + ZV( ) ( I A) xx w p xx o Here ϑ=θ α is relative roll angle; l(ϑ) is stability diagram. l( ϑ) h ϑ ϑ 3 = + ϑ 5 6 1, where h is initial metacentric height. Let us consider isolated heaving equation. Then we obtain the following expression ν + ω α δ = κ ( ω) ω w αw δw (5) [ ] [ ] Here the following denotations are adopted ν = D g µ 33 + λ 33 (4) γlb R, ω =, α =, δ = (6) 4 D B 3B + λ 33 g Analysis of equation (5) is simpler than (4). At the same time it allows to investigate main nonlinear effects related with low buoyancy of UV. 3. NUMERICAL MODELLING Analysis of system (4) presented in [] shows that interdependence of rolling and heaving is essential only in frequency interval corresponding to parametric resonance (ω /ω θ ). So we can consider only isolated heaving equation in form (5)-(6) for quality investigation of nonlinear effects stipulated by low buoyancy of UV. Equation (5)-(6) is nonlinear second order differential equation with constant coefficients. Since equations of such type admit exact solutions only on additional conditions on coefficients values [1] then most complete analysis could be carried out only numerically. 6th order Runge-Kutta method is used for numerical integration of (5). UV with length L=1 m, radius R=5.5 m is used for model calculations. Characteristics of different calculation variants are shown in table 1.

8 Table 1. Characteristics of UV position and stability at different drafts T. T Z p B D V λ 33 ν ω α δ % % % Nonlinear vertical oscillations at calm water and regular waves Nonlinear effect related with low buoyancy appears most obviously in free oscillations. Let initial condition be () = A; () = B (8) Phase portraits corresponding to free oscillations of UV when T=1 m (see tab.1) are shown in fig.4 for positive initial state A = 1. m and negative initial state A = -1. m. Essential differences in oscillations caused by elevation and sinkage of UV are observed from comparison of 4a and 4b. For conventional ship these two initial excitations are identical. From these figures we see that oscillations caused by elevation are smoother at first stage than those caused by sinkage. In the latter case the object passes first semicoil of phase portrait faster (fig.4b). Phase portrait in fig. 4a is more prolate than in 4b. Essential differences are observed only at first coil of phase portrait. At the next coils deviations are small, and restoring force is linear. One of the most important characteristics of forced regular oscillations of nonlinear system is relationship between oscillation amplitude and excitation frequency. Three amplitude characteristics for different UV reserve buoyancy (see initial data in tab.1) V={65%,%,7%} are shown in the fig.5. Contrary to conventional ship types, transfer function in heave has essential cavities. In other words there are such frequency ranges ω where waves have no influence on considered object. Width and place of this range are related with parameters of UV position relative water surface. Analysis of phase characteristic also shows that there are such oscillation conditions when UV heaves in opposition with wave, i.e. object does not heave on oncoming wave, but vice versa dives under it. It is stipulated by sign changing of reduction coefficient κ (ω). Moreover, calculations show that variation of UV reserve buoyancy is sufficiently effective means for heaving stabilisation. Theoretically we always can find value V in order that oscillations even on irregular waves could vanish. It is possible to observe phenomena of main and subharmonic resonance when minimum of transfer function and resonance zones do not coincide due to reduction coefficient. Phase portrait of UV oscillation near main resonance for V=5% is shown in fig.6a. This phase portrait is asymmetrical (prolonged on one side).this effect is also explained by dissymmetry of restoring moment (4) relative to current waterline. Phase portrait of UV oscillations in subharmonic resonance (ω ω /3) is shown in fig.6b. Oscillation shape essentially differs from fig 6a. Such asymmetric form was described by V.V.Lugovsky [1] for conventional ships too. 3.. Nonlinear heaving in irregular waves Distribution function and moments permitting estimate degree of nonlinearity process are interesting for investigation of nonlinear oscillations in irregular waves. These

9 characteristics are determined with the help of averaging one sufficiently long realisation of considered oscillations which are modelled by equations (5)-(6) with input process (15). The most important statistical characteristics of UV oscillation are mean value of heave (it characterises degree of heaving dissymmetry) and standard σ (it determines heave intensity). Let us consider third and forth moments (coefficients of skewness and kurtosis) for description of degree of nongaussity (i.e. nonlinearity) of UV motion. Estimations of these moments are A N 3 ( i ) E ( i 4 ) N = ; = 3 (8) 3 Nσ Nσ i= 1 i= 1 It is quite clear that, if original equation is linear and waves are gaussian process defined by autoregressive model (15), then distribution law of UV oscillations is gaussian, and coefficients of skewness and kurtosis are zero). Otherwise we can speak about deviation from normal distribution due to nonlinear properties of (4). Let us consider, for example, object motion with rather high reserve buoyancy V=16%, T=8.84 m, ω=.96 c -1, V=8185 t. Sea state is 5 number. JONSWAP approximation is used for wave spectrum. Model waves realisation with length 1 points was reproduced. Parameters of waves are the following: =. m., σ =.786 m., A =.394, E =.17. Element of realisation of UV motion in random sea is shown in fig.7. Histogram of heaving (t) distribution is shown in fig.8. Skewness and kurtosis values essentially exceeding boundaries of confidence intervals for zero hypothesis as well as shape of histogram distribution show that UV oscillations in irregular waves become essentially asymmetric. Surges ( sinkage ) occur during such motion. It is explained by lower level of reserve of potential energy for derivation upper current waterline than for the same derivation lower waterline. Influence of UV reserve buoyancy variation on degree of motion nonlinearity in random sea is shown in table. This influence is given by value Z p. Table contains statistical moments of (t) which are calculated basing on model data for above mentioned UV for irregular waves sea state 7. One can see from this table that draft increasing (reserve buoyancy decreasing) involves increasing of skewness and kurtosis coefficients. Thus influence of nonlinear factors is amplified. Besides nonzero deviation of middle waterline is appears during severe motion, i.e. UV on the average is in lifted position. Table. The UV draft dependence of statistical moments of UV heaving in irregular waves (sea state 7). Z p B σ A E

10 It is clear that obtained results demonstrate significant variation of UV motion from linear oscillations to dangerous side. Intensity of deep immersions of UV is essentially greater than predicted by linear model without regard to higher statistical moments. Nongaussian oscillation character even for traditional ship type could be caused by both influence of nonlinear terms in (4) and nongaussian exciting process, for example asymmetric waves. Therefore let us consider UV motion defined by equations (5)-(6) on asymmetric irregular waves defined by parabolic nonlinear transformation f(y)=ay+by for influence estimation of wave shape asymmetry on character of UV heaving. Object described in tab.1 with T=1 m, V=4.7% is used for calculations. Values of wave w moments and moments of UV heaving subject to nonlinearity index b are shown in tab.3 for sea state 5. Table 3 Moments of waves and heaving distribution of UV subject to nonlinearity index b. b σ A w E w σ A E One can see from tab.3 that increasing of nonlinearity index b results in increasing of mean waterline deviation. At the same time standard σ decreases. So asymmetric waves are less dangerous than symmetric ones. Coefficient of motion skewness increases when b increases. Coefficient of motion kurtosis depends inversely on b. Illustration of this fact is shown in fig.9 (b=. symmetric waves; b=. asymmetric waves). One can see that tops of UV oscillations ( emergence ) coincide whereas their hollows ( sinkage ) are significantly different. Motion amplitudes are essentially less than on symmetric waves. It is explained by the fact of more flat hollows of asymmetric waves (fig.). So sinkage amplitudes are less. At the same time emergence amplitudes are not significantly dependent on crest shape because UV reserve buoyancy is also insignificant. Therefore this example shows that asymmetric waves (dead swell) are less dangerous for UV with lower reserve buoyancy than symmetric waves (wind waves) to distinguish from, for example, small ships of traditional types. 4. DISCUSSION AND CONCLUSION Analysis of results of numerical modelling shows that heaving of UV with low buoyancy is essentially nonlinear due to significant decreasing of waterplane area with draft increasing. Such motion is asymmetric relative to UV calm waterline. Additional peculiarity of this effect is sign changing of reduction coefficient κ in range of wind wave numbers k. In nature it results in the following: 1. There is frequency range when exciting force acts in opposition with wave shape. It means that object dives under the wave instead of elevation with it ( as conventional ship).

11 . There is wave frequency range when exciting forces do not practically act on considered object. 3. Phase portraits and realisations of free decay heaving of UV caused by initial elevation or sinkage are essentially different due to asymmetry of restoring force. 4. Harmonic oscillations of UV with low buoyancy significantly differ from conventional ships motion. In particular phase portrait becomes asymmetric (prolonged) relative to zero line of heave and still symmetric relative to speed zero line. 5. Nonlinear effects of UV heaving are the same as for nonlinear rolling of conventional ships [13]: superharmonic resonance, bifurcation, deterministic chaos. In particular typical phase portrait of UV (see tab.1 for T=1 m) heaving is shown in fig Distribution law of UV heaving in irregular waves significantly differs from normal distribution. When reserve buoyancy decreases, then average deviation relative to calm waterline, skewness and kurtosis coefficients increase. 7. Influence of asymmetric waves on UV motion is more significant than that for conventional ship, but it involves reverse effect. Probability of large sinkage amplitudes decreases because hollows are more flat. At the same time it is necessary to note that we can consider these results only as qualitative. It is related, first of all, with simplified presentation of forces and moments acting on UV. In particular expressions for exciting force are obtained in the frames of Froude-Krylov assumption without diffraction component and with very simple damping law. Nevertheless, qualitative characteristics could be used for expert estimation during development of intelligence system (IS) knowledge base for monitoring of supermarine unsinkability. It permits to control emergence of UV in conditions of incomplete information. Such IS can operate in the following way (like []): The system turns on at the moment when UV moves under free surface at depth H (e.g. periscope depth for UV) and has heave motion due to wind waves. Parameters of wind waves on the surface (significant height, average period and wave direction) are estimated with the help of UV vertical acceleration record. Calculation of UV heaving in full surface condition is obtained. Possibility of safety emergence is determined. Block of optimal course choice for emergence begins operate if current safety emergence is impossible. Situation classification is performed if emergence is permitted. Model estimation of optimal reserve buoyancy is done if UV is subjected to strong motion on surface. This estimation is performed both from stability and heaving stabilisation point of view. In any case probability of dangerous accelerations and wetness frequency of UV is estimated. Emergence is performed.

12 REFERENCES 1. Lugovsky V.V. Nonlinear Problems of Seakeeping of Ship. Leningrad, Sudostroenie, 1966 (in Russian).. Belenky, V.L., Degtyarev A.B., Boukhanovsky A.V. Probabilistic Qualities of Nonlinear Stochastic Rolling. Ocean Engng, Vol. 5, No. 1, pp. 1-5, Spyrou K.J. Ship Capsize Assessment and Nonlinear Dynamics.// 4th International Ship Stability Workshop, St.John s, Newfoundland, September 1998, paper Spyrou K.J. Dynamic Instability in Quartering Seas: The Behaviour of a Ship During Broaching. Journal of Ship Research, 4, 1, March 1996, pp Thompson J.M.T. Nonlinear Mathematics and its Applications. Cambridge University Press, Cambridge, Kan M., Taguchi H. Chaos and Fractals in Nonlinear Rolling and Capsize of a Damaged Ship. //Proc. of International Workshop on the Problems of Physical and Mathematical Stability Modelling OTRADNOYE 93, Vol., paper, Kaliningrad Technical Institute, Kaliningrad, Tanizawa K., Naito S. An Application of Fully Nonlinear Numerical Wave Tank to the Study on Parametric and Chaotic Roll Motions. //Proc. of 6th International conference STAB 97, Varna, Bulgaria, September 1997, vol., pp Francescutto A. Nonlinear Ship Rolling in the Presence of Narrow Band Excitation. In: J.M.Falzarano and F.Papoulis (eds.), Nonlinear Dynamics of Marine Vehicles, ASME, New Jersey, 1993, pp Khaskind M.D. Hydrodynamic Theory of Ship Motion. Moskow, Nauka, 1973 (in Russian) 1. Newman J.N. Marine Hydrodynamics. The MIT Press, Cambridge, Blagoveschensky S.N. Theory of ship motion, New York, Dover Publications, Boroday I.K., Netsvetaev Y.A. Seakeeping of Ships. Leningrad, Sudostroenie, 198 (in Russian) 13. Lugovsky V.V. Mathematical Model for Investigation of Ship Non-linear Oscillation Stability in Waves.//Proc. of the International Symposium for Ship Hydrodynamics devoted to 85th anniversary of birthday of A.M.Basin, St.Petersburg, May 1995, pp (in Russian) 14. Longuet-Higgins M.S. The Statistical Analysis of a Random Moving Surface. Phil. Trans. Roy. Soc., London, 1957, 49, N966, pp Rozhkov V.A., Trapeznikov Yu.A. Probability Models of Oceanological Processes. Leningrad, Hydrometeoizdat, 199 (in Russian). 16. Boukhanovsky A.V., Degtyarev A.B. Probabilistic Modelling of Stormy Sea Fields.// Proc. of International Conference Navy and Shipbuilding Nowadays, St.Petersburg, February 1996, Vol., paper A-9. (in Russian) 17. Mandal S., Lyons G., Witz, J. Multivariate Autoregressive Algorithms for Ocean Wave Modelling. //Proc. of Second International Offshore and Polar Engineering Conference, San Francisco, USA, June 199, pp Olivera Pires H., Justino P., Pontes M. Autoregressive Representation of Random Sea Waves. //Proc. of Second International Offshore and Polar Engineering Conference, San Francisco, USA, June 199, pp

13 19. Boukhanovsky A., Divinsky B., Kosyan R., Lopatoukhin L., Abdalla S., Ozhan E. Short-term statistics by the measurements of the buoy near Russian coast of the Black sea. //Proc. of the International MEDCOAST Conference, April 1999, Antalya, Turkey, pp Boukhanovsky A.V., Degtyarev A.B. Nonlinear Stochastic Ship Motion Stability in Different Wave Regimes. //Trans. of 3rd International Conference CRF 96, St.Petersburg, June 1996, vol., pp Samsonov A.M. Travelling Wave Solutions for Nonlinear Dispersive Equations with Dissipation. Applicable Analysis, Vol. 57, pp Nechaev Yu.I., Degtyarev A.B., Boukhanovsky A.V. Analysis of Extremal Situations and Ship Dynamics in Seaway in an Intelligence /system of Ship Safety Monitoring. //Proc. of 6th International conference STAB 97, Varna, Bulgaria, September 1997, vol.1, pp κ (k) Fig. 1. Co-ordinate systems.5 65% 1% k 1/m Fig. Reduction coefficients when reserve buoyancy equal to 1% and 65%

14 Fig.3 Realisation of gaussian (solid line) and nongaussian (dash line) wave shapes..4 a) A = 1. m. b) A = -1. m Fig. 4. Phase portrait of free-decay oscillations Fig.5. Frequency dependence of heave motion of UV with different reserve buoyancy (dot line is 65%, dash line is %, solid line is 7%)

15 Fig.6a. Phase portrait of UV heave motion in main resonance mode. V=5% Fig.6b. Phase portrait of UV heave motion in superharmonic resonance mode. V=8%.

16 (t) Fig.7. Realisation of UV heaving on irregular waves (Sea State 5). P() Fig.8. Histogram of UV heaving on irregular waves (Sea State 5).

17 Fig.9. Realisation of UV heaving on symmetric (b=., solid line) and asymmetric (b=., dash line) waves t Fig.1. Phase portrait of UV heaving on regular waves with loop. V=1%