A STOCHASTIC MODEL OF RINGING

A STOCHASTIC MODEL OF RINGING Kurtis R. Gurley Department of Civil Engineering University of Florida Gainesville, Fl Ahsan Kareem Department of Civil Engineering and Geological Sciences University of Notre Dame Notre Dame, IN Federico Waisman EQE International San Francisco, CA Mircea Grigoriu Dept. of Civil and Env. Eng. Cornell University Ithaca, NY ABSTRACT A numerical study is conducted in which the ringing phenomenon observed in tension leg platforms is simulated on a single column piercing the water surface subjected to viscous wave forces. A statistical analysis of the input / output records is used to develop a probabilistic model to describe the observed ringing behavior. The wave forces used in the numerical simulation are the output of a non-linear filter subjected to non-gaussian colored noise generated using higher order spectral techniques. It is shown that these non-gaussian wave forces acting on the column can be represented by the sum of a translation and a Poisson white noise process. The statistics of the response of the oscillating column to this probabilistic model and the corresponding simulated records are in good agreement. Generation of wave forces with the probabilistic model greatly increases the efficiency of the response prediction, while retaining non-gaussian information in the input and output. INTRODUCTION Observations of the behavior of full-scale offshore platforms has revealed unexpected infrequent bursts of large magnitude, short duration response referred to as ringing. This high frequency transient type response has been observed in the vertical modes of tension leg platforms (TLPs), and is particularly troublesome for several reasons. This large amplitude behavior can significantly alter the fatigue life of platform substructure (e.g. TLP tendons), yet the simulation and analysis software used in design of these structures did not predict this behavior, despite attempts to include any relevant nonlinearities in loading and response. To date, the clearest conclusions are that the specific mechanisms leading to ringing response are not fully understood, making the incorporation of the ringing phenomenon in the reliability analysis of offshore platforms difficult. Ringing and springing are terms that are often interchanged, but describe very different modes of behavior. Figure demonstrates the distinction to be made in this study. Springing is a steady state response in the vertical and/or bending modes of TLPs and gravity based structures (GBS), often associated with second--order wave effects and vertical resonant response. This behavior is commonly observed in both mild and severe sea states. Ringing is the strong transient response observed under severe loading conditions. The transient response decays to steady state at a logarithmic rate that depends on the system damping. Ringing is a rare event, and has been unaccounted for in standard response analysis codes until recent experimental and full--scale observations brought them to light. It is presumed that ringing is triggered by the passage of a single high, steep wave event, or a wave train whose combined effects lead to very rapid build up of response to dissipate the input energy. This rapid build up is attributed to higher order effects in loading and structural response which occur only in rare circumstances. Ringing behavior is only observed under very specific nonlinear sea states and loading conditions combined with system parameters conducive to large energy build up under these circumstances. In recent years, significant efforts have been made toward the prediction of ringing. A small sampling of the studies attempting 5 x 3 Springing Ringing 5 5 5 Figure. Ringing and springing response in vertical mode

to identify the nonlinear mechanisms that induce ringing include (Davies 994, Jeffreys and Rainey 994, Natvig 994, Faltinsen 995, Newman 995, Gurley and Kareem 996). Studies reported in the literature have focused on large volume structures which are dominated by wave diffraction inertial type loading and minimally affected by drag forces (Davies 994, Jeffreys 994, Natvig 994, Faltinsen 995, Newman 995). These loads are calculated by a slender body theory or diffraction/radiation analysis. Full scale and model test observations (e.g., wave profiles and model behavior and validation of numerical procedures), are utilized in these studies. Natvig (994) details a number of mechanisms which contribute to ringing such as variable cylinder wetting and added mass, wave profile, wave slapping, and added mass. Using a model based on slender body theory with modified Wheeler stretching, Jeffreys and Rainey (994) show encouraging agreement between model tests and theory. Davies et al. (994) highlight the nature of loads that cause ringing based on model tests, present a time domain simulation scheme for estimating ringing response, and discuss simple guidelines for reducing the ringing response. Newman (995) and Faltinsen (995) discuss second and third--order sum frequency wave loads and their effects on ringing. Gurley and Kareem (996) have shown that viscous loads are also capable of inducing ringing response of members with large wavelength to diameter ratios. In that study, ringing response in pitching due to viscous loading is simulated on a simple pivoted cylinder which pierces the surface. The major contributing mechanisms are identified, and the system characteristics that influence the onset of ringing are delineated. Objectives The objectives of the current study are: () The simulation of long time histories of ringing response and the associated wave forces. () A statistical analysis to describe these records. (3) The development of a practical probabilistic model of the wave forces. (4) The evaluation of the accuracy of the probabilistic wave force model through comparison of the cylinder response subjected to wave forces produced by the probabilistic model, and the original input wave records. OVERVIEW, OSCILLATOR MODEL, INPUT GENERATION Let { Mt ()t, } represent the equivalent total moment acting on a cylindrical column piercing the water surface. This record is a nonlinear function of wave height elevation produced using standard techniques, with a second order correction added using higher order spectral methods. The moment time history is applied to an oscillator model and parameters are identified which induce ringing. Long realizations of Mt () are then generated and statistical descriptions are empirically derived. The wave force records from the simulation model are characterized by infrequent large values of very short duration that can be viewed as impulses of random magnitudes arriving at random times. These wave forces are modeled by the superposition of a non-gaussian translation process and a Poisson white noise. This two stage probabilistic model is used to simulate wave force input, and applied to the oscillator. Results are compared visually and statistically with the oscillator response to the original records simulated through spectral methods. Oscillator Model The JONSWAP wave elevation spectrum (Chakrabarti 987) is applied with a peak frequency of. Hz throughout this study. Stokes second-order random waves are simulated utilizing a quadratic transfer function (QTF) in the Volterra series framework. The QTF is analytically derived based on Stokes second-order random wave (Hasselmann 96, Hudspeth 979, Kareem et al. 994). Simulation details can be found in (Gurley et al. 996). The simulated realization of the surface elevation of gravity waves exhibit non-gaussian features with characteristic high peaks and shallow troughs. These non-symmetric profiles lead to skewed water particle velocity profiles, with higher probability of occurrence in the extreme tail region than in corresponding Gaussian waves. Application of the Morison equation to calculate the force at the mean water level ignores the nonlinear effects of the fluctuating free surface, which is thought to be a dominant ringing mechanism. The wave kinematics up to the instantaneous water surface are used to generate moment input from both linear and secondorder waves by integrating the force to the free surface and multiplying by an equivalent moment arm. The water particle velocity at the mean water level is related to the velocity profile on the wet- Ringing is simulated on an oscillating column piercing the water surface, as shown in Fig.. The wave force acting on the oscillator is an equivalent total moment resulting from integration of the wave force from the bottom of the submerged column to the instantaneous free surface. The pivoted cylindrical column of diameter D oscillates about a fixed center of rotation, at a distance c r below the mean water level. The draft, d r, is the column length below the mean water level, and wave elevation is positive above the mean water level. The depth d is set at, meters. The equation of pitching motion is described by mθ () t + cθ () t + kθ() t = Mt () where m is the system moment of inertia, c is the system damping, k is the system stiffness, θ, θ, θ are the rotational displacement, velocity and acceleration, respectively. Mt () is the moment due to hydrodynamic loads for which a probabilistic model is sought. The fluctuating moment of inertia and associated damping and stiffness are not considered in the analysis since the changes are small compared to the given system parameters. D cr + dr η() t Figure. Cylinder model in wave train Wave Input Model and Equivalent Moment Calculation d ()

ted cylinder using modified Airy stretching theory (Mo and Moan 985). The time dependent moments acting on the cylinder are calculated from the water particle velocity using the Morison drag term, M = LF l R, where, F l = ρ C d Du u is the viscous force per unit length on the cylinder, ρ, C d and u are the fluid density, coefficient of drag, and water particle velocity, L is the fluctuating wetted cylinder length, and R is the fluctuating equivalent moment arm. In this study the column is very small with respect to wavelength in order to isolate viscous effects, thus the Morison inertia term is not considered The total applied wave moment is a combination of four components, where M = --ρc d Dη η -- cr umwl u mwl na M = --ρc, d Du i u i dl dl ( na i)dl + ---- (3) i = dl = ( n cr) na, n =, η >, η, η nb M 3 = -- ρc d Du i u i dl ( i )dl + dl ---- i = dl = ( dr + cr) nb nb M 4 -- ρc d Du i u i dl ( i )dl dl = + ---- η i = dl = ( dr + η) nb In the above equations, u mwl is the water particle velocity at the mean water level. The cylinder below the mean water level is discretized to calculate the local force per unit length due to the exponentially decaying velocity profile. The portion above the mean water level is divided into na parts, and its moment contribution is given above as M. The portion below is divided into nb parts, and its moment contributions are given by M 3 and M 4. The local water particle velocity at the i th discrete portion of the cylinder is u i, and d l is the length of the discretized section. There are three combinations of the four moment expressions in Eqs. -5 for three different combinations of the instantaneous wave elevation η with respect to the center of rotation and the mean water level, η >, η > cr ==> M = M + M + M 3, (6) η <, η > cr ==> M = M + M 3, (7) η <, η < cr ==> M = M 4. (8) () (4) (5) Sample Result: Nonlinear Wave Effects An example of a simulated wave train with and without non-linear correction, and the resulting responses are shown in Fig. 3. A Gaussian wave field and the cylinder response are seen in graphs and. The same wave field with the second-order contribution added and the resulting response is in graphs 3 and 4. In this case, clearly the nonlinear wave input triggers ringing while the linear wave input does not. The response to nonlinear waves is positively skewed due to the skewness in water particle velocity, and has a high kurtosis due to the ringing events. Both the skewness and kurtosis lead to problems associated with extreme response and fatigue of ocean systems. It is noted that not all large waves in the non-gaussian wave train lead to ringing. Water particle velocity is a function of frequency and wave elevation, and it is found that steeper waves lead to ringing due to a quick build up of energy in the form of higher water particle velocity. 5 graph 5 5 5 5 3 5 linear input wave elevation response to linear input, cr= 3.5 draft=.. graph 5 5 5 3 nonlinear input wave elevation graph 3 5 5 5 5 3 response to nonlinear input, cr= 3.5 draft=.. graph 4 5 5 5 3 Figure 3. Gaussian waves and cylinder response (graphs,), non-gaussian waves and cylinder response (graphs PROBABILISTIC MODELS OF WAVE FORCES Let M = { M i, i =,,...,n} be a time series giving values of Mt () at equal time intervals t >. Figure 4 shows a realization of this time series with n = 5, and t = sec. The series was simulated as a nonlinear function of second order wave elevation, and is characterized by large infrequent values of short duration, referred to as impulses. These impulses cause the ringing phenomenon observed in the response. It is proposed to separate these impulses from the underlying continuous input history and model the two components separately. Let M c be a new time series derived from M by removing the impulses present in the wave force record. The series M c is called the continuous component of M in this study. The remain-

.5 x 4.9.8.5.7.6.5.4.5.3...5.5.5.5 3 3.5 4 4.5 5 x 4 8 6 4 4 6 8 x Figure 4. Realization of Mt () ing portion of M is the impulse behavior, denoted as M i. The two components are assumed uncorrelated and independent, giving the total moment a simple superposition of the two Mt () = M c () t + M i () t Eleven independent realizations of M of the same duration and time step as the record of Fig. 4 are used to estimate the marginal distribution and the spectral density function of M c and the statistics of the impulses in M i. Independent realizations of the non- Gaussian continuous and impulsive components are then generated using their respective models and superimposed. Marginal distribution of M c Let F m be the empirical distribution of the continuous component M c of the wave force process. Figure 5 shows with a solid line this empirical distribution estimated from the set of eleven independent realizations of M after filtering out the impulsive component. The model (9) Figure 5. Marginal distribution of M c () t where { Zt ()t, } is a stationary Gaussian process with mean zero and spectral density g () f given by, g () f g o ( f f o ) = -------------------------------------------------------------------- ( ( f f o ) ) + ( ζ o f f o ) () with parameters g o = 388.8, f =.98 Hz., and ζ o =.565. The covariance function of Yt () is EZt [ ( + τ)zt ()] so that the power spectral density of Yt () µ y is given in Eq.. Figure 6 shows with a solid line the average power spectral density g of the eleven independent realizations of M c. The figure shows that a significant fraction of the wave force energy is concentrated in the frequency range (.6,.4) Hz. Because the impulses in M occur very infrequently, the differences between the power spectral density of Fig. 6 and the one corresponding to the entire record M including the impulses are negligible. F m =.5.5( exp( a ( x + b ))), x b.46.4( exp( a x) ), b x.5 + (.5) exp ( a x), x b 3.7 +.3( exp( a 3 ( x b 3 ))), x b 3 () 8 of F m with parameters a =.3, a =., a 3 =.46, b =, and b 3 = 8 is shown with a broken line in Fig 5. The agreement between the empirical distribution and the model of Eq. is satisfactory. Spectral density of M c Let Yt () be a stationary Gaussian process with mean µ y defined by Yt () = µ y + Zt (), t () 6 4.5..5..5.3.35.4.45.5 freq (Hz) Figure 6. PSD of Mt ()(solid), Yt ()(dotted), and X c () t (broken)

Translation Model of M c The statistics and empirical marginal probability density function associated with the continuous component M c show a significant departure from Gaussian. The stationary Gaussian process Yt () from the previous section can represent the continuous component of the wave force process after transformation to non- Gaussian using a translation process X c Yt () µ () t F m Φ y = --------------------- = σ y qyt ( ()) (3) where σ y denotes the standard deviation of Yt () (Grigoriu, 995). The marginal distribution F m is given in Eq.. The mean and correlation function of this process are EX [ c () t ] = EqYt [ ( ())] EX [ c ()X t c ( s) ] = EqYt [ ( ())qys ( ( ))] (4) The translation process suggested in Eq. 3 will distort the power spectrum of Yt (), as seen by comparing the PSD of Yt () and X c () t (broke line) in Fig. 6. Therefore, the spectral model parameters following Eq. were selected such that the distortion renders a process X c that matches that of the empirically measured continuous component. Details of this method can be found in (Grigoriu, 998). The resulting model X c matches the marginal distribution and the correlation structure of the continuous component M c of the wave force process. Therefore, M c can be approximated by X c, and Eq. 9 then becomes Mt () = X c () t + M i () t (5) Two asides concerning X c are worth mentioning here. The definition of the continuous component X c of the wave force process can be modified by replacing the models F m and g with the estimates F and g, respectively. However, the use of the models F m and g is more convenient for numerical calculations and they are used in the paper. Impulse Model It is assumed that the impulses of the wave force process arrive in time according to a Poisson process. The validity of this assumption is verified by considering the time series realizations of Mt (). Figure 7(a) shows with a solid line the empirical distribution F τ of the interarrival times between consecutive impulses, where τ i, i =,...,9. The broken line in Fig. 7(a) is the distribution of an exponentially distributed random variable with parameter λ = E[ τ]. The resulting average interarrival time between consecutive impulses E[ τ] is 5,688 sec., so that the estimate of the mean impulse arrival rate is λ =.76. The agreement between the empirical data and the Poisson assumption model is satisfactory. In addition to arrival times, the magnitude and shape of the impulse must be modeled. For simplicity, the impulse shape is represented by the delta function, and has no duration. The magnitude (a).8.6.4. 3 4 interarrival x 4 is sufficiently modeled using the following Weibull distribution, obtained using 9 impulses F I ( x) x = exp ------------------- c, (6) α x where α = 3883.79 and c =, and is shown as the broken line in Fig. 7(b). Figure 7(b) also shows with a solid line the empirical distribution F i of the impulse magnitudes estimated from the set of eleven independent realizations of Mt (). The distributions have the same fourth central moment. The agreement between the empirical distribution and the model in Eq. 6 is satisfactory. Probabilistic model of M The statistical analysis of the wave force data suggests that the process where (b) X c Yt () µ () t F m Φ y = ---------------------, σ y 3 impulse magnitude x 4 (7) (8) can be used to approximate Mt (). The first term of the definition of X is the model X c of Eqs. and 3, and corresponds to the continuous component M c of M. The second term X i of X describes the impulses of the wave force process and is modeled by a Poisson white noise (Grigoriu, 995). The impulse component X i depends on a homogeneous Poisson counting process Nt () with mean arrival rate λ, the independent random variables { Y k } following the distribution F i, the delta Dirac function δ, and the random times { τ k } corresponding to the jumps of N. The impulses of the model X of the wave force process are assumed to have no duration. The model of Eq. 7 can be generalized to generate impulses of any specified shape. This extension can be achieved by replacing the Poisson white noise in Eq. 8 by a fil-.8.6.4. Figure 7. Impulse model for M Xt () = X c () t + X i () t Nt () X i () t = Y k δ( t τ k ) k =

tered Poisson process. Numerical results in this paper are for the wave force model of Eqs. 7, 8. RESPONSE ANALYSIS Consider the single degree of freedom linear system in Eq. with parameters m = 876.88,, c = 8.73, and k = 873.3. Figure 8(a) shows the system response θ() t subjected to the wave force record shown in Fig. 4, generated using higher order spectral techniques. Let θ be response of the oscillator to wave forces generated using the probabilistic model, represented by the solution of mθ () t + cθ () t + kθ () t = Xt () (9) where X is given by Eqs. 7, 8. Let θ c and θ i be the stationary responses of the linear system of Eq. 9 to X c and X i, respectively. Then, the response to the wave force model X is θ = θ c + θ i. Figure 8(b) shows the system response θ() t of Eq. 9 to a sample of the wave force process Xt () = X c () t + X i () t. Figure 9 shows in more detail the ringing phenomena in the response θ of Eq. 9. This figure details the response of Fig. 8(b) around time t =, sec. The transient response associated with a ringing event can be observed clearly and resembles Fig.. The decay rate depends on the system damping. The response analysis of Eq. 9 is performed in three steps. First, the theory of filtered Poisson processes is used to find ana- lytically the response component θ i () t to the input impulses X i () t defined by Eq. 8. Second, the Monte Carlo simulation method is used to calculate realizations of the response component θ c () t to the input X c () t defined by Eq. 8. Third, the total response θ as shown in Fig. 9 is defined by the superposition θ = θ c + θ i. Figure shows the power spectral densities of θ and θ. The solid line is the average power spectral density of the response θ of the system in Eq. to the eleven independent realizations of Mt (). The broken line in Fig. is the power spectral density of θ of the response of Eq. 9. The power spectral densities have two distinct peaks. These peaks are located at. and.4 Hz and correspond to the wave force energy content and the oscillator natural frequencies, respectively. The agreement between the response of the system in Eq. to the empirical data and the proposed model is satisfactory. Figure shows with a solid line the marginal distribution of the solution θ of Eq.. The marginal distribution of the response θ of the linear system to the proposed model is shown with a broken line in Fig.. The agreement is satisfactory. Finally, table shows that the stationary first four moments of the system responses θ and θ are in very good agreement. In particular, the higher order statistics which show large deviation from the Gaussian are recreated quite well by the probabilistic wave model. Table : Stationary Response moments (a) (b) statistics θ θ mean -8.34e-4 8.4e-4 variance..9 skewness.677.74 4 x 4 4 x 4 kurtosis 88.98 99.89 Figure 8. System response to simulated wave trains using (a) the original second order wave samples, (b) Equation 7..4.3. 3.5 4 x 4 3..5...3.4.5.94.95.96.97.98.99 x 4 Figure 9. Observed ringing in θ.5.5.5..5..5.3.35.4.45.5 freq. (Hz) Figure. PSD of θ and θ

.9.8.7.6.5.4.3....5..5.5..5. x Figure. Cumulative distribution function of θ and θ CONCLUSIONS Non-Gaussian wave trains are simulated using higher-order spectral methods, and applied as input to a single degree of freedom oscillator. The system parameters are set such that ringingtype behavior is induced infrequently. The equivalent total moment resulting from the integration of the wave force is non- Gaussian with infrequent impulses. This instantaneous moment is modeled as two independent components consisting of a continuous, non-gaussian part, and an impulsive part. Models of each component are introduced through use of a translational model and a Poisson white noise, where model parameters are estimated from the original records. This probabilistic model is applied to generate realizations of force input to the oscillator, and its response to this input is compared with the response to the original records generated using spectral methods. Results showed good agreement, suggesting that the probabilistic model is an efficient means of generating impulsive non-gaussian ringing behavior. ACKNOWLEDGEMENT The support for this work was provided in part by an ONR Grant N4--93----76. REFERENCES Chakrabarti, S., 987, Hydrodynamics of Offshore Structures, Springer-Verlag, Boston. Davies, K.B., Leverette, S.J., and Spillane, M.W., 994, Ringing of TLP and GBS Platforms, BOSS -Behavior of Offshore Structures, McGraw Hill Faltinsen, O.M., Newman, J.N., and Vinje, T., 995, Nonlinear wave loads on a Slender Vertical Cylinder, Journal of Fluid Mechanics, 89, 79-98. Grigoriu, M., 995, Applied Non-Gaussian Processes: Examples, Theory, Simulation, Linear Random Vibration, and MATLAB Solutions, Prentice Hall, Englewood Cliffs, NJ. Grigoriu, M., 998, Simulation of Stationary Non-Gaussian Translation Processes, Journal of Engineering Mechanics, ASCE. 4(), -7. Gurley, K., Waisman, F., Grigoriu, M., and Kareem, K., 997, Probabilistic Models of Ringing, Proceedings of the 7th International Conference on Structural Safety and Reliability [ICOS- SAR], A.A. Balkema Press, Kyoto, Japan, Nov., 997. Gurley, K. and Kareem, A., 996, Numerical Experiments in Ringing of Offshore Systems Under Viscous Loads, 5th International Conference on Offshore Mechanics and Arctic Engineering, ASME, June 6 -, Florence, Italy. Gurley, K., Kareem A. and Tognarelli, M., 996, Simulation of a Class of Non-Normal Processes, International Journal of Nonlinear Mechanics, Elsevier, 3(5), 6-67, 996. Hasselmann, K., 96 On the nonlinear energy transfer in a gravity wave spectrum, part I, Journal of fluid mechanics., 48-5. Hudspeth, R.T. and Chen, M., 979, Digital Simulation of Nonlinear Random Waves, Journal of the Waterway Port Coastal and Ocean Division, ASCE, 5(), 67-85. Jefferys, E.R., and Rainey, R.C.T., 994, Slender Body Models of TLP and GBS Ringing, BOSS. Kareem, A., 995, The Next Generation of Tuned Liquid Dampers, Proc. of the First World Conference on Structural Control, Los Angeles. Kareem, A., Hsieh, C.C., Tognarelli, M.A., 994, Response analysis of offshore systems to nonlinear random waves part I: wave field characteristics, Proceedings of the special symposium on stochastic dynamics and reliability of nonlinear ocean systems, Ibrahim and Lin (eds.), ASME, Chicago, IL. Mo, O., and Moan, T., 985, Environmental Load Effect Analysis of Guyed Towers, Journal of Energy Resources Technology, ASME, 7, 4-33. Natvig, B.J., 994, A Proposed Ringing Analysis Model for Higher Order Tether Response, Proceedings of the Fourth International Offshore and Polar Engineering Conference, Osaka, Japan, 4-5. Newman, J.N., 995, To Second Order and Beyond, The Texas Section of the Society of Naval Architects and Marine Engineers TLP Technology Symposium, unpublished proceedings. Stansberg, C.T., 993, Non-Gaussian Properties of Second-Order Sum Frequency Responses in Irregular Waves: A numerical Study, OMAE, Offshore Technology, ASME,, 48-489.