Roll dynamics of a ship sailing in large amplitude head waves

J Eng Math DOI 10.1007/s10665-014-9687-4 Roll dynamics of a ship sailing in large amplitude head waves E. F. G. van Daalen M. Gunsing J. Grasman J. Remmert Received: 8 January 2013 / Accepted: 18 January 2014 Springer Science+Business Media Dordrecht 2014 Abstract Some ship types may show significant rolling when sailing in large-amplitude (near) head waves. The dynamics of the ship are such that the roll motion is affected by the elevation of the encountering waves. If the natural roll period (without forcing) is about half the period of the forcing by the waves, then a stationary solution will have an amplitude that is much larger than for other forcing frequencies. This phenomenon is called parametric resonance. For certain hull shape types the transverse stability may vary considerably due to the waves passing the ship. Moreover, near head waves will also have a direct effect on the roll dynamics. For these processes a differential equation model a Mathieu type of equation is formulated. Furthermore, the waves considered are of a type that is encountered in open seas. As a parameterization of these waves the Pierson Moskowitz spectrum is used. The risk that the ship will reach a critical state is characterized by the time of arrival at this state, starting from an arbitrary pattern of the waves and the dynamic state of the vessel in the stationary situation. Large-scale Monte Carlo simulations of this process are carried out. The percentiles of the arrival time distribution indicate the risk of significant rolling to which the vessel is exposed. Furthermore, a method is proposed to estimate the maximum roll angle in a stationary state by taking into consideration only the part of the wave spectrum that relates to the state of parametric resonance. The result is compared with the outcome of the large-scale Monte Carlo simulations. Keywords Critical roll amplitude Parametric resonance Stochastic waves E. F. G. van Daalen M. Gunsing Maritime Research Institute Netherlands (MARIN), Wageningen, The Netherlands J. Grasman (B) Mathematical and Statistical Methods Group, Wageningen University and Research Centre, Wageningen, The Netherlands e-mail: johan.grasman@wur.nl J. Remmert Department of Maritime Technology, Delft University of Technology, Delft, The Netherlands Present Address: J. Remmert Maersk Maritime Technology, Copenhagen, Denmark

E. F. G. van Daalen et al. 1 Introduction In October 1998 a post-panamax C11 class cargo ship sailed on the Pacific Ocean from Taiwan to Seattle. While traversing a heavy storm, the vessel began an extreme rolling motion with roll angles of up to 40 to each side. After the storm had settled, the crew examined the status of the cargo and found that one-third of the deck-stowed containers were lost, and another third was heavily damaged. This incident is the greatest container casualty known so far; see [1] for a detailed account of the events. The ship experienced a phenomenon known as parametric resonance; within only a few roll cycles, the roll angle reached a value far above what would be considered normal (mostly up to 10 ). This kind of ship behaviour was known from the 1950s, but it was considered to be relevant only for smaller vessels in following seas. Since the October 1998 incident, interest has been renewed. It has been suggested [2] that the specific hull shape of modern container ships might increase the risk of parametric roll. To enlarge the load capacity while keeping the calm water resistance small, the length and width of container ships increased, and wide, flat sterns and pronounced bow flares appeared. This had a negative effect on dynamic stability in waves. Some possible countermeasures to avoid heavy rolling are the installation of (a combination of) bilge keels, stabilizing fins and anti-roll tanks of passive (free surface) or active (U-shape) type. Bilge keels are frequently applied but do not always provide sufficient additional roll damping. The effectiveness of fin stabilizers decreases with decreasing speed and increases the overall fuel consumption. Anti-roll tanks have the disadvantage of taking up a significant amount of space, thereby reducing the cargo capacity. That is why the last two measures are not applied to modern cargo ships. The estimation of the risk of parametric resonance for a given ship geometry and loading condition is of great importance to the maritime transport industry. In the literature this risk is usually quantified by a single coefficient in the equation of motion the roll damping coefficient. A critical value for the roll damping coefficient is derived based on a comparison between roll damping from the hull and the one that occurs in an unstable situation; see [3 5]. Here we take the stationary solution for a given resonant forcing as our starting point. We assume that near the resonance frequency the roll amplitude changes linearly with the forcing amplitude, so that with the relevant part of the wave spectrum an estimate of the roll amplitude can be made. The accuracy of this approximation method is analysed by computing the time until the critical state is reached for the full wave spectrum, starting from a probable state of the waves and the roll angle. From the distribution of these arrival times obtained from large-scale Monte Carlo simulations we calculate the appropriate percentiles for quantifying the risk that the critical state will be reached. In Sect. 2 we consider the way in which a single-frequency wave affects the roll amplitude of the stationary solution for a small range of frequencies around the parametrically resonant state. In Sect. 3 we introduce the Pierson Moskowitz wave spectrum and develop an efficient discretization method based on the inverse of the cumulative wave spectral density function. In Sect. 4 the roll amplitude is approximated from the part of the wave spectrum that corresponds with the parametrically resonant state. Large-scale Monte Carlo simulations are presented in Sect. 5 for different values of the two parameters of the Pierson Moskowitz wave spectrum as they may hold in open seas. For different seas and given arrival times at the critical state, the risk of attaining this critical state within these times is computed. Finally, in Sect. 6 the results are evaluated and the consequences of making some restricting assumptions are discussed. 2 Parametric resonance in the roll dynamics of a vessel In this section we analyse the roll dynamics of a vessel forced by a (near) head wave with a single frequency. To that end, we formulate the following one-degree-of-freedom differential equation model for the roll angle φ of a ship: (I + A) φ + B( φ) φ + Cφ = M exc, (1a)

Roll dynamics of a ship sailing in large amplitude head waves where the dot indicates differentiation with respect to time t. This equation describes a mass spring system with non-linear damping and external forcing; it is a generally accepted model for the roll motion of a ship in waves. By considering the roll motion only, we exclude interaction effects due to the coupling of roll motion with surge, sway and yaw motions. In doing so, we adopt the approach followed in the existing literature. However, the effects due to heave and pitch motions are included in our model, as will be explained subsequently. In (1a), I = 2.0 10 10 kg m 2 is the moment of inertia and A = 3.8 10 9 kg m 2 the added mass coefficient taken at the natural frequency ω nat of the vessel, depending on the forward speed U. The vessel-specific data are based on a C11 class container ship. The damping coefficient consists of a linear and a quadratic part: B( φ) = b 1 + b 2 φ, where b 1 = 1.6 10 8 Nm/(rad/s) and b 2 = 5.7 10 9 Nm/(rad 2 /s 2 ) are the linear and quadratic damping coefficients depending on U, respectively. These values are obtained from a roll decay experiment at 6 kn forward speed. Since the roll motion is typically narrow-banded (i.e. concentrated around the natural frequency of roll), the damping effect due to radiated waves is represented quite well by (1b). For a typically wide-banded response, memory effects should be accounted for by calculating the convolution integral of the retardation function and the velocity history. The restoring coefficient C consists of a mean part C 0 = 1.4 10 9 Nm/rad and a time-varying part δc: C = C 0 + δc = ρgv(gm 0 + δgm), where ρ is the seawater density (taken as 1,025 kg/m 3 ), g the acceleration of gravity (taken as 9.81 m/s 2 ), V = 74 10 3 m 3 the displacement volume and GM 0 and δgm the mean and time-varying parts of the (transverse) metacentric height, respectively. The linear form (1c) of the roll restoring coefficient is acceptable for a small to moderate roll angle regime, which applies to our analysis. For higher roll angles a more accurate representation of the GZ curve is required. The (undamped) natural frequency is calculated from ω nat = C 0 I + A(ω nat ). If we assume deep water conditions, then for a ship with forward velocity U and a wave with frequency ω approaching the ship at an angle μ = 180 (head waves), we obtain for the encounter frequency ω enc = ω + Uω2 g. (3) Compared to the other terms in (1a) the wave excitation roll moment M exc is small in (near) head sea conditions, and its effect is negligible. Therefore, for the remainder of this analysis we will assume that M exc = 0. The remaining forcing term δgm in (1c) is brought about by an incoming (undisturbed) wave with height = A cos(ωt + β ), where A is the wave amplitude and β the wave phase. The forcing term is described as follows: δgm = A δgm A cos(ω enc t + β + β δgm ), (5) where A δgm and β δgm are the amplitude ratio and the phase of the metacentric height variation. Thus, the differential equation describing the roll motion is given by (I + A) φ + ( b 1 + b 2 φ ) φ + C 0 φ = ρgvδgmφ. (6) This means that the way the wave acts upon the vessel is expressed in a transfer function denoting the manner in which the wave amplitude and phase are linearly transformed in the forcing function on the right-hand side of (6). This linear transfer function is obtained from quasi-static considerations of the instantaneous wave height along the moving ship. Figure 1 shows the amplitude A δgm and phase β δgm (with respect to the wave height) of the metacentric height variation for a wave amplitude of A (lin) = 1m. (1b) (1c) (2) (4)

E. F. G. van Daalen et al. Fig. 1 Linear transfer function of metacentric height variation: amplitude (solid line) and phase (dashed line). For an arbitrary value of the wave frequency ω the amplitude change and phase shift are generated by linear interpolation on the complex amplitude Fig. 2 Parametric resonance indicated by squared roll amplitude of stationary solution of (1) (7) as a function of the encounter frequency ω enc for A = 1 m. Numerical results ( ) are based on the transfer function values given for the parametrically resonant case. The constant approximation (dashed line) is chosen for an interval with a length such that the enclosed surface equals the surface under the computed (solid) line. From a full wave spectrum the part that induces parametric resonance is identified by the length ω enc = 0.046 rad/s of this interval In Fig. 2 the roll amplitude for the stationary solution is given for the encounter frequency interval [0.42, 0.54] rad/s, with U = 6.0kn= 3.1 m/s and A (lin) = 1 m. Furthermore, the transfer coefficients A δgm and β δgm are fixed at the value that corresponds with ω enc = 0.48 rad/s. Note that near this value the amplitude rises considerably from the periodic behaviour of the restoring coefficient C; seeeq.(1c). Actually, then the encounter frequency ω enc is approximately twice the natural frequency: ω enc 2ω nat. The roll motion amplitude A φ shows a peak (resonance) value of A φ,res = 0.39 rad 2 = 0.63 rad = 36, while away from the resonance frequency the amplitude is very small. In the literature this behaviour is described mathematically by a pendulum of variable length [6]. Using the preceding result for A (lin) = 1 m and assuming that the functional relation A φ = A φ (A ) is close to linear satisfying A φ (0) = 0, a wave with a frequency ω res = 0.42 rad/s and amplitude A brings about a roll motion with a frequency close to twice the natural frequency ω nat = 0.24 rad/s and with amplitude (7) (8) A φ = A φ,res A A (lin). (9) 3 Waves represented by a Pierson Moskowitz spectrum In the previous section we considered the effect of a single-frequency wave on roll dynamics. In reality sea waves are composed of waves with different frequencies. The distribution of these frequencies depends on the geographical data and the weather conditions and is given as a spectral density function S. Two parameters play an essential

Roll dynamics of a ship sailing in large amplitude head waves role in the specification of S : the peak wave period T p and the significant wave height H s. The significant wave height can be expressed directly in the zeroth-order moment m 0 of the spectrum H s = 4 m 0 with m 0 = 0 S (ω) dω. We choose the two-parameter Bretschneider or Pierson Moskowitz wave spectrum [7], which is a frequently used type of spectrum for fully developed seas: S (ω) = a ( ω 5 exp b ) ω 4, (11) where the spectrum parameters a and b are related to the peak wave period and the significant wave height as follows: ( ) 4 1/4 a T p = 2π 5 b and H s = 2 b. (12) To apply an efficient discretization of the spectrum, we define the cumulative spectral density function Q (ω) def = ω 0 S (ω ) dω = a 4b exp ( b ω 4 ), (13) with inverse ( ) Q 1 b 1/4 (q) =. (14) ln (a) ln (4bq) We split up the spectrum into N ω intervals with the areas under the curve at an interval all having the value d(a, b) = m 0 = a. (15) N ω 4bN ω To that end we define the median of each interval by (( def ω j = Q 1 j 1 ) ) ( 1/4 a b = )) 2 4bN ω ln (N ω ) ln ( j 2 1, j = 1, 2,...,N ω. (16) In Fig. 3 it is shown how the discretization works out for N ω = 10. The points separating the intervals are the set of expected values if nine points chosen from a statistical distribution with the shape of the spectrum [8]. In practice, much larger values for N ω are chosen. The advantage of this approach is that for some required approximation accuracy the number N ω is smaller than for a method that uses equidistant points: since all wave components represent the same area d(a, b), all points contribute equally to the total energy. A wave composed in the aforementioned way of N ω components takes the form N ω N ω ( ) = j = 2d(a, b) cos ω j t + β j, (17) j=1 j=1 where the phases β j have randomly chosen values with a uniform distribution on the interval [0, 2π]. (10) 4 Critical sea states based on stationary roll amplitude As a starting point we consider the numerical stationary solution of Eqs. (1a) (1c) forced by a sea parameterized by a Pierson Moskowitz wave spectrum. For parametric resonance only the part of the wave spectrum near the resonance frequency ω res = 0.42 rad/s is important. In Fig. 2 we replaced the numerically computed (solid) line by

E. F. G. van Daalen et al. Fig. 3 Pierson Moskowitz wave spectral density S, with H s = 5.5m and T p = 14.5 s. The area under the spectral density curve is divided into N ω = 10 equal parts of size d(a, b);see(14) (15) Fig. 4 Critical significant wave height H s,crit as function of peak wave period T p for different values of critical roll amplitude A φ,crit for numerical stationary solution of Eqs. (1a) (1c) a (dashed) line with the same enclosed area. It covers an interval of length ω enc = 0.046 rad/s on the encounter frequency axis. Since from (3) we can derive that dω enc dω = 1 + 2Uω g, it is concluded that from the spectrum S (ω) only an interval of length / dωenc ω res = ω enc dω (ω res) = 0.036 rad/s, (19) with ω res = 0.42 rad/s as centre, must be considered. Consequently, we take for the resonant wave as amplitude the value A = 2S (ω res ) ω res. (20) Now, from Eqs. (9), (11) (12) and (20) we can easily establish a relation between the peak wave period T p,the critical significant wave height H s,crit and the critical resonance roll amplitude A φ,crit : H s,crit ( Tp ; A φ,crit ) = A φ,crit T 2 p π 2 A φ,res ω 5 res 10 ω res exp (18) ( ( ) ) π 4 10. (21) ω res T p In Fig. 4 curves are given above which the wave spectrum parameters have values that yield a roll amplitude larger than a given critical value in a stationary state (e.g. for A φ,crit = 36, see dashed curve). Thus, based on the numerical stationary solution of Eqs. (1a) (1c) and on the estimates of the amplitude we made, we may conclude which seas are safe (below the curve) and unsafe (above the curve). In the next section we deal with the case where the wave pattern is not stationary because of random fluctuations. Then we must consider the time-dependent problem of reaching a critical state given an arbitrary starting state in a safe phase. 5 Distribution of arrival times at a critical roll energy It is worthwhile considering the energy balance as part of the roll dynamics. Energy enters the system through the incoming wave and dissipates through damping due to wave radiation and friction. At a given instance the total energy E tot stored in the roll motion equals the sum of kinetic energy E kin and potential energy E pot :

Roll dynamics of a ship sailing in large amplitude head waves E tot = E kin + E pot = 1 2 (I + A) φ 2 + 1 2 C 0φ 2. (22) This study aims at quantifying the risk of parametric roll. As a measure for this risk one could take the time until a certain critical value ±φ crit is reached, given some initial state of the roll angle and the wave. This would be a rather sensitive indicator because, given a certain amount of energy stored in the roll motion, it may happen that for a very short moment this threshold is only slightly exceeded. For this reason we consider the time until the total energy reaches the value corresponding to the potential energy at φ = φ crit : E tot = E crit def = 1 2 C 0φ 2 crit. (23) For a sea state characterized by the values of the parameters a and b of the Pierson Moskowitz spectrum (11) we compute the arrival time distribution if the threshold value E crit is reached within a finite time. Since integration over large time intervals may be needed, we must take into account that in fact the wave height given by (17) is quasi-periodic. To overcome this problem, we change the wave input by making it a stochastic process. In (17) the phases β j are given fixed random values. A better representation of the random effects that continuously influence the waves is obtained by letting β j be a Brownian motion process: dβ j = εdw (t), where dw (t) denotes the standard Wiener increment [9] and ε is a phase modulation parameter. If the (quasi-) periodicity of the input (17) is felt at a time scale T, then for ε one should take ε = T 1. A realization of β j (t) is obtained from a forward Euler scheme with a sufficiently small time step t: β j (t + t) = β j (t) + εr(t) t. Here, r(t) is a random number having a standard normal distribution with zero mean and unit variance. It is remarked that this white noise component changes the spectrum and adds energy to the system: for all frequencies S (ω) is increased by ε 2. Consequently, wε 2 /m 0 (with w the bandwidth) gives an indication of the relative size of the effect. Since w/m 0 is of the order O (1), the effect is of the same order as ε 2 itself. The numerical simulations are based on Eqs. (1a) (1c), which are rewritten as a set of first-order, non-linear differential equations with initial conditions d dt ( φ φ ) = ( φ ( B ( φ ) φ + Cφ ) /(I + A) ), ( ) φ (0) = φ (0) (24) (25) ( ) 0. (26) 0 These equations are solved with an explicit first-order Euler forward scheme. The restoring coefficient C is evaluated at each time step taking into account Eqs. (5) and (24), with phase modulations applied to all frequency components. For all possible combinations of the peak wave period and the significant wave height, the solution is calculated for a long time (Fig. 5). Depending on ( T p, H s ),see(12), the system will show one of the three following types of behaviour: 1. If the excitation is very weak, then the system will remain in the safe zone (where E tot < E crit ) at all times. 2. If the excitation is very strong, then the system will go directly into the unsafe zone (where E tot > E crit ) and it will remain there for the rest of the simulation. 3. For all cases in between very weak and very strong excitation, the system may enter and exit the unsafe zone from time to time. We register the consecutive time stages at which the system enters (at t = t in ) and exits (at t = t out ) the safe zone. From these entry and exit times, we calculate the lengths of the time intervals during which the system is in the safe zone: T int = t out t in. Next we consider an infinitely large set of Monte Carlo runs as follows: one large run is made for which the integration is continued regardless whether the system is in the safe or unsafe zone (Fig. 5a). Then we consider all starting points t 0 within a safe zone i forming the interval [t in,i, t out,i ]. This approach guarantees that the system will startatanytimet 0 in a state that corresponds with a stationary situation given the parameters of the wave spectrum.

E. F. G. van Daalen et al. (a) (b) Fig. 5 Schematization of Monte Carlo simulation procedure to determine distribution of exit (arrival) times. In a safe zone the total energy is below the critical energy E crit, meaning the maximum angle also stays below φ crit. a A run in which the critical energy is exceeded repeatedly. b Typical example of output from a single Monte Carlo simulation run to determine the distribution of exit (arrival) times with T p = 16.0s and H s = 6.5 m. The critical energy level E crit is indicated by the dashed line. Left: full simulation run (up to approximately t = 85 h) with 1,000 exits from the safe zone. Right: excerpt from simulation (from t = 50 h to t = 51 h) displaying a few exits from the safe zone Since each starting value has an equal probability, we can compute the fraction q i (τ) of runs that will arrive within a given time τ at the critical energy E crit as follows: ( ) q i (τ) def τ = min, 1. (27) T int,i Next we take a weighted average of the first j safe zones within the large run: Q j (τ) def = j T int,i q i (τ) i=1. j T int,i i=1 (28)

Roll dynamics of a ship sailing in large amplitude head waves Fig. 6 Estimating the risk of reaching the critical energy before time τ = 10 min with T p = 16.0 sandh s = 6.5 m based on the first j safe intervals with j = 1, 2,...,1,000 using (27) for Q j (τ) and the unbiased result S j (τ) based on one randomly chosen start value at each safe interval. The wave spectrum is discretized with N ω = 50 and ε = 0.01 Fig. 7 Output from numerical simulations: probability of an exit time less than 10 min as function of peak wave period T p and significant wave height H s. Dashed curve: boundary between safe sea states (lower area) and unsafe sea states (upper area) based on stationary approach described in Sect. 4 with critical angle amplitude of 36 Simulation runs starting in the same safe interval are statistically interdependent because they are exposed to the same stochastic wave pattern during the time they overlap. Consequently, Q j (τ) may give a biased estimate of the risk of reaching a critical energy within time τ. Drawing only one starting value from a safe interval based on a uniform distribution over this interval and weighting the scores over all safe intervals in the same way as for Q j (τ) gives an unbiased estimate S j (τ). However, in the latter case of single runs, the convergence is slow for increasing j (Fig. 6). It is also noted that the limit value of Q j (τ) stays close to the expected limit value of the unbiased sequence S j (τ), the fraction of single runs up to safe interval j that reach the critical energy within timeτ. In Fig. 7, the fraction Q j (τ) is presented for j = 1,000 and τ = 10 min and for a variety of combinations of the peak wave period T p and the significant wave height H s using 50 frequencies in the discretized wave spectrum. The position and shape of the level contours match very well with the result from the stationary approach presented in Sect. 4. In a sense, the stationary result can be seen as the 100 % limit solution from the non-stationary approach. 6 Conclusions and recommendations The aim of this study was to quantify the risk of significant rolling in (near) head waves containing a frequency component which is twice the natural roll frequency of the vessel. The resulting large roll response is known as parametric resonance and was mathematically analysed in the prototype differential equation known as the Mathieu equation [10]. The starting point of our study on roll dynamics was the numerical solution of the one-degreeof-freedom equations of motion [Eqs. (1a) (1c)] near this resonance frequency with given parametric forcing amplitude. It led to a numerical approximation of the roll amplitude as a function of the frequency (Fig. 2). Next we considered the composition of sea waves as parameterized by the Pierson Moskowitz spectrum. Using the result given in Fig. 2 we were in a position to identify the part of the spectrum which could be linked to the resonance state and led to the resonance forcing amplitude given by (20). With this formula the resulting roll amplitude was easily derived for different values of the peak period and significant wave height (Fig. 4). The outcome compared quite well with that of a stochastic stability analysis of roll dynamics introduced in [4]; see Fig. 11 in [5]. Our rather straightforward approximation method can be improved in different ways, as follows:

E. F. G. van Daalen et al. In (1a) the restoring moment is approximated by a term proportional to the roll angle φ. We also verified that using the exact formula with sin φ does not change the results significantly. Using sin φ instead of φ does not require much additional effort, but then a comparison with the existing literature could not be made that easily because mostly the linear approximation is used. The same argument applies to the use of the linear roll restoring coefficient (1c) (Sect. 2). The expression for the roll amplitude, being linear in the forcing amplitude (9), can be replaced by e.g. a quadratic function. Then in addition to the case A = 1 m a second numerical solution is needed within the range that corresponds to realistic roll amplitudes (e.g. A = 1.25 m). In the Monte Carlo simulation we considered the full Pierson Moskowitz spectrum discretized in a number of frequencies covering the spectrum in an efficient way. Furthermore, instead of working with the roll amplitude in the stationary state for the resonant wave frequency we considered the evolution of the roll amplitude when starting with a probable state of the vessel in a given sea. The total energy in the roll motion is considered to be a good measure to distinguish between safe and unsafe states. In risk analysis the occurrence of a catastrophic event should be well defined and include the time span that is taken into consideration. Thus, the probability should be found that a certain critical value will be reached before a given time for all feasible values of the parameters T p and H s of the wave spectrum. In Fig. 7 these probabilities are given for the event that the energy will exceed a given critical level within 10 min. The result is obtained from Monte Carlo simulations based on an integration of Eqs. (1a) (1c). In our study we applied a mixed analytical numerical approach in which we computed the resonant state itself and used it in our risk analysis, while in the existing literature [1 6] the destabilization of the rest state of (1a) (1c) is analysed by estimating which forcing would lower the damping coefficient B critically. In fact, this is a local (near equilibrium) analysis which becomes less accurate for larger forcing. Acknowledgments The authors would like to thank the organizers and participants of the Study Group Mathematics with Industry for initiating a first exploration of the subject; see [11]. References 1. France WN, Levadou M, Treakle TW, Paulling JR, Michel RK, Moore C (2003) An investigation of head-sea parametric rolling and its influence on container lashing systems. Mar Technol SNAME News 40(1):1 19 2. Santos-Neves MA, Rodriguez CA (2007) Influence of non-linearities on the limit of stability of ships rolling in head seas. Ocean Eng 34:1618 1630 3. Dunwoody AB (1989) Roll of a ship in Astern Seas Metacentric height spectra. J Ship Res 33:221 228 4. Dunwoody AB (1989) Roll of a ship in Astern Seas Response to GM fluctuations. J Ship Res 33:284 290 5. Levadou ML, Van t Veer R (2006) Parametric roll and ship design. In: Proceedings of the 9th international conference on stability of ships and ocean vehicles, vol 1, pp 191 206 6. Shin YS, Belenky VL, Pauling JR, Weems KM, Lin WM (2004) Criteria for parametric roll of large container ships in longitudinal seas. Trans Soc Naval Archit Mar Eng 112:14 47 7. Pierson WJ, Moskowitz L (1964) A proposed spectral form for fully developed wind seas based on the similarity theory of S.A. Kitaigorodskij. J Geophys Res 69:5181 5190 8. David HA, Nagaraja HN (2003) Order statistics, Wiley Interscience, Hoboken 9. Gardiner CW (1990) Handbook of stochastic methods for physics, chemistry and the natural sciences. Springer, Berlin 10. Tondl A, Ruijgrok T, Verhulst F, Nabergoj R (2000) Autoparametric resonance in mechanical systems. Cambridge University Press, Cambridge 11. Archer C, Van Daalen EFG, Dobberschütz S, Godeau M-F, Grasman J, Gunsing M, Muskulus M, Pischanskyy A, Wakker M (2009) Dynamical models of extreme rolling of vessels in head waves. In: Molenaar J, Keesman K, Van Opheusden J, Doeswijk T (eds) Proceedings of the 67th European Study Group Mathematics with Industry, pp 1 27. ISBN 978-90-8585-600-9