Proceedings of the ASME 27th International Conference on Offshore Mechanics and Arctic Engineering OMAE2008 June 15-20, 2008, Estoril, Portugal

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1 Proceedings of the ASME 7th International Conference on Offshore Mechanics and Arctic Engineering OMAE8 June 15-, 8, Estoril, Portugal OMAE DISTRIBUTION OF MAXIMA OF NON-LINEAR ROLLING IN CASE OF COUPLED SWAY AND ROLL MOTIONS OF A FLOATING BODY IN IRREGULAR WAVES Ignace D. Mendoume Minko Marc Prevosto Marc Le Boulluec Ifremer Ifremer Ifremer Centre de Brest Centre de Brest Centre de Brest ABSTRACT The so-called Linearize & Match L&M, which gives a good approximation of the exact distribution of maxima roll angle of non-linear systems, was studied some years ago by Armand and Duthoit 199 and by Prevosto 1. The developments within this method were made in the case of single degree of freedom dynamic systems. Moreover, the terms mass, damping, stiffness of the non-linear transfer function did not depend on the circular frequency. In this paper, first, the L&M method is improved by adding a last step in the procedure which correct the Gaussian closure technique of the method, secondly is generalized to a coupled sway and roll dynamic system in which the hydrodynamic coefficients are frequency-dependent. The system is modelled by a set of ordinary differential equations in which the non linearity is only in the roll motion. In order to validate the results obtained in this case by the L&M method, a Monte Carlo method with long simulations of the response of the system was carried out. Hence, some aspects of the time domain simulation, based on Cummins equations, are also discussed. INTRODUCTION Prediction of floating offshore structures and ships motions is of growing interest since it is directly related to the safe and economics issues of ships and offshore platforms. This problem has been extensively studied in the frequency domain using transfer functions. However, this approach is only of interest in the case of linear systems. For nonlinear systems, time domain simulations have became a common approach to analyse such systems. Nevertheless, this technique is usually computationally expensive. Consequently, less time consuming stochastic procedures, such as techniques based on the theory of Markov processes and the associated Fokker-Planck equation, equivalent linearization, have been developed. These methods however, are based on assumptions of weak damping or of broadband input spectrum. This last assumption is generally incompatible with wind sea spectra situations and even less with swell conditions. More recently methods using the first-order reliability method FORM Jensen, 7 have been studied but have not been yet compared to the method presented in this paper. A method called Linearize & Match L&M Armand and Duthoit, 199 and Prevosto, 1, allows to avoid the above restrictions, and furnishes very good approximation of the exact distribution of maxima. The method first consists on approximation of the original system subjected to a Morison-type nonlinearity by an equivalent nonlinear system in which the nonlinearity is replaced by a cubic polynomial function. The parameters of the cubic system are determined by minimizing, in mean square sense, the difference between the original system and the cubic system. Afterwards, higher-order statistics of response are derived from a sequence of equivalent linear systems, whose parameters are chosen in order to provide an accurate approximation of high order moments of the original system, using Volterra series expansion or perturbation techniques. Once obtained the moments, a maximum entropy method is used to obtain the marginal densities of roll and of its derivative. Hence, the joint density of roll and its derivative is approximated using a method of perturbation Monbet et al, Finally, using Rice formula the extremes statistics of roll response are deduced. The development and validation of the L&M methods have been done in the case of single degree of freedom SDOF dynamic systems. Moreover, the coefficients from radiation and diffraction forces were not frequency-dependent. In this paper, the L&M method is generalized to a coupled sway and roll dynamic system in which the hydrodynamic coefficients are frequency-dependent. Moreover, a last step in the 1 Copyright 8 by ASME

2 procedure has been added which correct the Gaussian closure technique used to obtained the variance of the system. This is done by using in the closure the marginal distributions obtained by the L&M method instead of the Gaussian one. This improved method is so called in the sequel Linearize & Match & Iterate L&M&I. Comparisons are also made with results from Monte Carlo simulations. Some aspects of the time domain simulation are also presented in this paper. EXTREME ROLLING ANALYSIS Sway and roll motions The governing equation of motion considered in this study to describe the coupled sway and roll motions of a floating body, is of the form Ÿt Ẏt [I p + I aω] + B rω + θ t θ t B q θt θ t Yt +[K h + K a] = F h ω 1 where I p is the matrix of inertia, I aω and B rω are the frequency-dependent added mass matrix and linear wave damping matrix, respectively, B q is the nonlinear damping coefficient, K h the hydrostatic stiffness matrix, K a the mooring stiffness matrix, Y and θ are the sway and roll response and F h the incident waves forces acting on the floating structure. One can observe that, in the Eq. 1, the nonlinearity is only in the roll motion. As shown in Fig. 1, the excitation i.e. F h, is considered as a simple linear transfer, the so-called RAO, applied to the free surface, ηt, which is obtained as a linear filtered Gaussian white noise, bt. In this case the excitation and the free surface elevation are Gaussian processes. Probability law of the maximum rolling Let Θ c be the random variable representing the positive maxima roll angle. It was shown by Rychlik and Leadbetter that, in the case of a narrow-banded roll response, the probability that Θ c will exceed a certain level u is equivalent to the ratio between the average number of upcrossings of this level, denoted N u +, and the average number of zero-upcrossings, denoted N +. Mathematically, this distribution is PΘ c > u = 1 F Θc u = u θ t f Θc ydy IE[N+ u ] IE[N + ] where IE stands for the expectation value and F Θc u is the probability distribution function of Θ c. The average number of upcrossings of a level u, in the stationary case, is obtained by Rice formula Rice, 1954 as follows IE[N + u ] = zf θ, θ u, zdz 3 We note that this expression requires the joint probability density function of rolling and its derivative, f θ, θ.,., which is difficult to obtain for an arbitrary non-gaussian process. For a standardized Gaussian stationary process with its second spectral moment m finite, a simple integration of Eq. 3 gives IE[N u + ] = m1/ π e u 4 Some other methods use an approximation of the joint distribution of the roll angle and its derivative. In this paper the joint density is approximated by a polynomial perturbation of the product of marginal densities f θ, θu, z = f θ uf θzsu, z 5 with su, z obtained by the projection method as described by Monbet et al The main idea behind the projection method is to approximate the function su, z by a polynomial function [ su, ] z which preserves the value of the cross-moments m ij = IE θ i θj for any i + j L, for a given power degree L. A comprehensive description of this technique can be found in Monbet et al The calculation of marginal densities f θ u, f θz and crossmoments necessary to estimate su, z is carried out by the L&M&I method associated with a Maximum Entropy Method MEM. LINEARIZE & MATCH & ITERATE METHOD The L&M method has shown a good promise for the approximation of the distribution of maxima of scalar nonlinear systems. In this section, we show how this method can be generalized in the case of a coupled sway and roll system with frequency-dependent coefficients. Hereafter, we give the different steps of the L&M&I method. For a comprehensive discussion one can refer to the works of Armand and Duthoit 199 and Prevosto 1. Equivalent cubic system The L&M method as described by Armand and Duthoit 199 and Prevosto 1., is not directly applicable to the type of Eq. 1. The non-linearity is approximated by a cubic plus linear damping. We will thus work with the following system Ÿt Ẏt [I p + I aω] + B rω + θ t θ t α θ t + β θ t 3 Yt + [K h + K a] = F h ω 6 where the coefficients α and β are obtained so that IE [ ɛ ], with ɛ = B q θt θ t α θ t + β θ t 3 is as small as possible in the mean square sense. This leads to the following expression for α and β, based on a Gaussian closure α = B q IE[ θ ] π ; β = B q θ t 9πIE[ θ ] Copyright 8 by ASME

3 Hydrodynamic linear filter Mechanic nonlinear filter bt h bη t ; H bη ω ηt h ηfh t; H ηfh ω F h t h Fh Xt; H Fh Xω Xt Waves Forces Response Fig. 1 Schematic of the system For convenience, let set M ω = I p + I aω ; B ω = B rω ; K = K h + K a X t = Yt θ t Ẏ ; Ẋt 3 3 = t θ 3 t Then, Eq. 6 can be rewritten as ; I 1 = 1 1 M ω Ẍ t + B ω + αi 1 Ẋt + βi1ẋ3 t + KX t = F h ω 11 Sequence of equivalent linear systems In a second stage one considers a sequence of equivalent linear systems M ω Ẍ it + B ω + αi 1 + βγ ii 1IE [Ẋ i t] Ẋit +KX it = F h ω 1 whose matrix coefficients Γ i defined by [ ] γ 11 i γi 1 Γ i = γi 1 γi 13 are determined such that : IE [ X i] IE [ ] Xi i. That is, coefficients Γ i are calculated so that the ith order moments of the response X it approximate as accurately as possible the ith order moments of the response Xt of the system Eq. 6. The Γ i are obtained with a procedure which is described in Armand and Duthoit 199. Without any loss of generality, Eq. 1, due to the construction procedure, can be rewritten as follows [ ] M ω Ẍ it + B ω + αi 1 + βγi IE θ i t I 1 Ẋit +KX it = F h ω 14 [ ] That is, we set : γi 11 = γi 1 = γi 1 = and IE θ i t = ] IE [Ẏi θi =. In this case one shows that γ J i = 3 + i 1 IE[θlin ]IE[ θ 15 lin ]I where J = h t τrθ 3 lin θ lin dτ I = h t τr θlin dτ θlin 16 with h an element of the impulse response matrix function and θ lin solution of the linear system Ÿlin t M ω + [B ω + αi 1] θ lin t Ẏlin t + K θ lin t Ylin t = F θ lin t h ω17 Roll maxima probability law Once the γi calculated, the moment of second order of the roll and its derivative are approximately obtained from the sequence of linear systems Eq. 14. And the higher moments are classically obtained using the Gaussian properties of the linear process solution of Eq. 14. Knowing the higher moments, the marginal densities of the roll and its derivative are obtained by a maximum entropy method. The joint density will then be calculated from Eq. 5. Correction step Linearize & Match & Iterate L&M&I In order to improve the results, final stage is added, which consists in correcting the variances by modifying the equivalent linear system. For that we use the distribution of θ obtained after the maximum entropy method step to correct the closure technique. The methodology is given here for a simple 1 dof system but could be used without problem to the system studied in this paper Eq. 6. Classically the equivalent linear system is obtained as a quadratic minimization of the residue between the non-linear damping term and the equivalent linear one. Let the system be I θ + T θ θ + Kθ = F h t 18 and the equivalent linear system I θ + α L θ + Kθ = Fh t 19 We look for the minimum of IE[ɛ ], with ɛ given by ɛ = T θ θ α L θ 3 Copyright 8 by ASME

4 The solution is easily obtained as α LIE[ θ ] = TIE[ θ θ ] 1 In the Gaussian case the normalized expectation of θ θ is IE[ θ θ ] IE[ θ ] 3/ = 4 π So the linear coefficient with the Gaussian closure is taken as α L = T 4 π IE[ θ ] 1/ 3 Now if we use the distribution of θ, obtained with the L&M method, to compute numerically the expectation of θ θ and of θ IE[ θ θ ] L&M = IE[ θ ] L&M = The new linear coefficient is taken t t f θtdt 4 t f θtdt 5 α L = IE[ θ θ ] L&M IE[ θ IE[ θ ] 1/ = aie[ θ ] 1/ 6 ] 3/ L&M and furnishes a better linear equivalent system, with the new variances obtained as previously in L&M method in iteratively solving with IE[ θ ] = ω Hω S Fh F h ωdω 7 1 Hω = Iω + jωaie[ θ ] 1/ + K and S Fh F h the spectral density of the input F h. TIME DOMAIN SIMULATION Equation of motion in the time domain 8 This section investigates the motion response of a floating body in time domain. Because Eq. 1 is nonlinear it is necessary to formulate the equations of motions in the time domain. According Cummins 196 and Ogilvie 1964 works, Eq. 1 in the time domain can be written as [I p + I a ] Ẍt + t Rt τẋτdτ + B q θt θ t + [K h + K a] X t = F h t 9 where X t = Y t, θ t T, I a is the hydrodynamic added mass coefficient evaluated at the infinite frequency limit and Rt is called the memory function or the retardation function. These parameters are related to the frequency-dependent terms in Eq. 1 in the following form Rt = π I a = I aω + 1 ω B rω cosωtdω 3 Rt sinωtdt 31 Hence, if the values of the linear damping or added mass are known for circular frequencies, from zero to infinity, one can, using the above relations, deduce the retardation function. However, in practice, damping and added mass are only known for a limited interval of frequencies. So, one problem in the numerical computation of Eqs. 3 and 31 is how to estimate the values of B rω and I aω at lower i.e. as ω tends to zero and higher frequencies. In this paper Brω and I aω, for lower and higher frequencies are approximated by polynomial functions. Time series of wave exciting forces As we can see, Eq. 9 requires as inputs time series of the wave exciting forces. Let F h ω, and Aω be the Fourier transform of F h t and ηt, respectively. Then we have F h ω = H ηfh ωaω 3 were H ηfh ω is the response amplitude operator RAO between wave force and wave amplitude. It results that time series of the exciting force are given by : F h t = F 1 F h ω, where F 1 is the inverse Fourier transform. In this study, the RAO is given from the hydrodynamic software DIODORER. The only unknown parameter in Eq. 3 is the Fourier transform of ηt, that is Aω. In the following we show how to obtain it. Since ηt is a stationary and Gaussian process, it is well known that the incident wave elevation can be defined by N ηt = Re A qe jωqt+φq q=1 33 with N sufficiently large and where A q = S ηηω + q ω, with S ηηω + q the one-sided spectrum of the process ηt and {φ q, q IN} a sequence of independent and identically distributed random variables with uniform distribution in [, π]. In practice it is common to use the complete spectral density, S ηηω q, defined by S + ηηω q = { Sηηω q if ω q otherwise and Eq. 33 can be rewritten as ηt = N q= N 34 Aω qe jωqt 35 with Aω q = ω/s ηηω qe jφq. Finally, the wave forces are determined as follow F h t = N q= N H ηfh ω q S ηηω q ω/e jωqt+φq 36 4 Copyright 8 by ASME

5 Or F h t = Re N q=1 H ηfh ω q S ηηω + q ωe jωqt+φq 37 The results of the time domain simulation are time series for which statistical properties can be derived. Hence, the empirical estimator, that we will consider as the reference, of the average number of upcrossings, is of the form IE [N u] = 1 N t N 1I ],u] ]u,+ ] θt k, θt k + t 38 k=1 with t the time step and N the number of time steps. NUMERICAL STUDIES In this section a model of FPSO is investigated for numerical applications. Results from frequency domain, that is matrix coefficients and response amplitude operator RAO, have been obtained by the hydrodynamic analysis tool DIODORER. The analysis was performed in time intervals ; the range chosen for this analysis is [4, 3]second, with time step t = 1s and headings ranging from to 36 degree with step β = 5deg. Due to the limited time intervals of analysis made by DIODORER, the natural period range chosen for the following analysis is T = [; 35]s. asymptotic fittings using polynomial functions are applied for added mass and damping coefficients. The results are shown in Figs. and 3. The corresponding memory functions are shown in Figs. 4. One can observe that the memory effects in the response vanish after about 4s. Time simulation The time domain simulation is performed with a wave record derived from an unidimensional spectrum obtained by integrating a directional spectrum which is the product of a JONSWAP spectrum Hasselmann et al, 198 with significant wave height Hs = 1m, the spectral peak wave period Tp = 15.3s and the spectral shape γ = 1 and a spreading function with parameters β = 9deg and s = see Fig. 5a. The peak period of the JONSWAP spectrum is close to the roll resonance frequency. From these values, the times series of the free surface elevation and corresponding forces are derived see Figs. 5b and 6, respectively. Finally, the nonlinear system Eq. 9 is solved in uniform time steps using the fourth order Runge-Kutta method. The time step is t = 1s and the duration of simulation is T = 18s. The memory functions are computed before the routine which solves the motions equations is called. It was found that the solution of the sway motion is very sensitive to sudden variations of the exciting force, which induce low frequency sway motion associated to the mooring stiffness. Thus a sinusoidal ramp function was adopted to initiate the motion. This process secures that the structure will not encounter waves with large amplitudes from equilibrium position immediately after simulation starts. Figs. 7 represent time series motions for sway and roll motion. Hence the average number of upcrossings, used to calculate the law of maximum see Eq., is calculated through the formula Eq. 38. Comparative study The comparaison between the L&M, the L&M&I and the Monte Carlo simulation are based on the variances, the normalized higher moments and the law of distribution of peaks. Tab. 1 shows the statistical moments which are normalized, apart for second order moments, as follows [ ] [ IE θ k θl / IE [ θ ] [ ] ] k/ l/ IE θ 39 Fig. 8 shows the distribution probability of rolling obtained by the two methods. One can observe that L&M&I method gives very good agreement with simulation estimates and show well the non-gaussian properties of the response process which is taken into account in the L&M&I method. The differences between the variances given by the L&M and L&M&I methods show all the interest of the last step of iteration with non Gaussian closure. The values obtained for a Gaussian process are given into parentheses. Ia4 [kg.m] Ia [kg] Br [kg/s] Br4 [kg.m/s] 3 x x x Ia4 [kg.m] 1 5 x Ia44 [kg.m ] 6 x 11 Fig. Interpolated added mass x Br4 [kg.m/s] Br44 [kg.m /s] x x 19 Fig. 3 Interpolated damping Copyright 8 by ASME

6 R 1 x x 18 R4 1 1 x x 18 Sway displacement [m] Time[s] R Time [sec] R Time [sec] Fig. 4 Memory functions for sway and roll Roll angle [ ] Fig. 7 Time series motions of sway and roll S η η f [m] ηt [m] Frequency [s] Probability distribution function Roll [ ] Fig. 5 Unidirectional wave spectrum and time series of free surface elevation F 1 t [N] F t [N.m] 4 x x Fig. 6 Time series of excitation forces Fig. 8 Distribution of maxima of roll. L&M&I blue, Simulation Black CONCLUSIONS The Linearize and Match method developed by Armand and Duthoit, improved here in this paper in the Linearize and Match and Iterate L&M&I method to predict the response statistics of a scalar non linear stochastic process has been extended in the case of a two degrees of freedom system, for the determination of the distribution of coupled sway and roll. A time domain method has been presented which solves the coupled nonlinear system using the fourth order Runge Kutta method. Results from simulation are then used to provide statistical moments, roll maxima distribution which are compared to the corresponding from L&M&I method. These two method have been tested on a example of roll response of a FPSO. The results obtained show the accuracy of the L&M&I method. ACKNOWLEDGEMENTS The authors would like to thank Principia, who provided data and for the permission to publish the results of the present study. 6 Copyright 8 by ASME

7 [ ] IE θ k θl l = l = l = 4 l = 6 l = 8 normalized SIM LMI SIM LMI SIM LMI SIM LMI SIM LMI k = LM k = LM k = k = k = Table 1 Comparison of the moments. SIM : obtained by Monte Carlo simulation, LMI : obtained by Linearize&Match&Iterate methodgaussian process, LM : obtained by Linearize&Match method REFERENCES [1] Jensen, J. J., 7, Efficient Estimation of Extreme Non- Linear Roll Motions Using the First-Order Reliability Method FORM, J. Mar.Technol., 1, pp [] Armand, J. -L., and Duthoit, C., 199, Distribution of Maxima of Non-Linear Ship Rolling, nd Intl. Symposium on Dynamics of Marine Vehicles and Structures in Waves, Procs. Publ. by Elsevier Sciences Pubs., pp [3] Prevosto, M., 1, Distribution of Maxima of Non-Linear Barge Rolling with Medium Damping, Proc. ISOPE Conf., vol. III, pp [4] Monbet, V., Prevosto, M., and Deshayes, J., 1996, Joint PDF Parametric Models for Rate of Upcrossings of the Acceleration for Non-Linear Stochastic Oscillator, Int. J. Non-Linear Mechanics, vol. 31, no. 5, pp [5] Rychlik, I., and Leadbetter, M.,, Analysis of Ocean Waves by Crossing Oscillation Intensities, International Journal of Offshore and Polar Engineering, 1, pp [6] Rice, S. O., 1954, Mathematical Analysis of Random Noise, BellSystems Technical Journal, 3, pp. 8-33, and 4, pp [7] Cummins, W. E., 196, The Impulse Response Function and Ship Motions, Schiffstecnik B.D., 47, 9, pp [8] Ogilvie, T. F., 1964, Recent Progress Towards the Understanding and Prediction of Ship Motions, In Proc. 5 th Symposium of Naval Hydrodynamics Bergen, [9] Hasselmann, D. E., Dunckel, M., and Ewing, J. A., 198, Directional Wave Spectra Observed During JONSWAP 1973, Physical Oceanography, 1, pp Copyright 8 by ASME