Identification of Quadratic Responses of Floating Structures in Waves

Identification of Quadratic Responses of Floating Structures in Waves im Bunnik, René Huijsmans Maritime Research Institute Netherlands Yashiro Namba National Maritime Research Institute of Japan ABSRAC A formulation for the estimation of the low frequency damping and the quadratic wave drift force transfer function will be presented in this paper. Synthesized time series of the waves and low frequency motions are analyzed. Parameters influencing the stochastical nature of the estimation of the quadratic transfer functions, such as length of time series, number of time segments and frequency resolution, will be discussed. From the simulated low frequency damping and the quadratic transfer function the input signals for the analysis will be reconstructed as a means for a quality check. In this paper two estimation methods will be discussed. One method based on cross-bi-spectral analysis and a method which is based on a minimization scheme involving the quadratic transfer coefficients and the reconstruction of the second order time series. KEY WORDS System Identification non linear processes, Model test analysis. INRODUCION In the day to day practice of performing model tests on moored floating structures in waves one is often faced with the question of the validation of computed motion or force RAO's against the measured ones. Especially for floating structures where non-linear effects can play a significant role, such as semi-submersibles in survival wave conditions. Since the beginning of the 6 s analysis procedures for non-linear identification techniques have been developed (see ick (96)). hese techniques were based on a volterra series expansion of the kernels. he analysis of non-linear processes has been put in a more rigorous mathematical frame work by Brillinger (97), where he introduces the description of poly spectra and higher order transfer functions. A deconvolution technique making it possible to separate linear and higher order responses has been demonstrated by Bendat (99). Later Dalzell (97,975) used cross-bi(tri)-spectral techniques, based on cross-bi-covariance estimators to determine the mean added resistance transfer function and the cubic transfer function for roll of vessels sailing in waves. he quadratic wave drift force transfer function is normally not measured directly. Only in special model test programs such as e.g. reported by Pinkster (98) and Huijsmans et al (99). he quadratic transfer function is estimated directly from force measurements. Stansberg (994, 997) has shown that using a de-convolution process also from low frequency motion responses the underlying quadratic transfer function and the low frequency damping coefficients can be established. In this paper we will focus on the practicality of the developed procedures as well as the statistical relevance of the estimated parameters. Mathematical Background We will shortly explain the basics of linear and non-linear responses. Assume we have a Gaussian distributed input process x(t) with zero mean and a finite r.m.s in a time segment from to. We can describe this as: iπυt x(t) = X ( υ)e dυ, tε(,] () S XX,( υ ) = E X ( υ ) () he linear response to such an input x(t) follows from: () iπυt y(t) = H ( υ)x ( υ)e dυ. (3) From which the cross spectrum can be derived as: S YX,( υ ) = E Y ( υ)x ( υ ) (4) he averaging is taken by splitting the time series into a number of segments. For a quadratic response formally written as: () () y(t) = y (t) + y (t) (5) Paper No. 6-JSC-435 Bunnik

Or alternatively: () () i πυ υ ( )t y (t) = H ( υ, υ )X ( υ )X ( υ )e dυdυ (6) Where H is the quadratic (second order) transfer function. A more common way of notation is when we apply a rotation in the integration variables giving: () iπµ t () y (t) = e dµ H ( υ, υ+µ )X ( υ )X ( υ+µ )dυ (7) In the following section we will describe two methods for the estimation of the wave drift force transfer function. Method Cross-bi-spectral analysis Fourier transform of the previous equation gives us: ( ) ( ) * ( ) ( ) ( ) (( )) Y ω = dωh ω, ω X ω X ω ω=ω ω (8) While the left hand side comes from: Y dt y t y e π ω ( ω ) = { () } i t m he quadratic transfer function satisfies the symmetric properties shown as follows: ( ) ( ( ) ) ( )* ( )* H ω, ω = H ( ω, ω ) = H ( ω, ω ) = H ( ω, ω ) () he cross-bi-spectrum between x() t and y() t can be written as: π S, lim E X X Y * ( ω ω ) = ( ω ) ( ω ) ( ω) (( ω=ω ω )) xxy (9) () Where E represents an ensemble average and means the measured time. We can obtain the following expression that allows us to estimate QF: ( ω ω ) ( ω ) S ( ω ) ( ) S, H ( ω, ω ) = S xxy xx xx () Note that we deal with two-sided QF here. If we use the one-sided expression, the right hand side of Eq. should be divided by a factor. Eq. means that we can estimate QF with Eq. and the spectra of the input signal. In the case that the output signal is equal to the squared input signal, QF should be and we can obtain the relation: ( ) ( ) ( ) S ω, ω = S ω S ω (3) xxx xx xx Using this strict relationship we can calibrate the quadratic transfer function by scaling the directly estimated transfer function with the transfer function for the squared operator. Discretization Here we divide the measured time traces of wave and force into M sections respectively to prepare for ensemble averaging. Discretizing Eq. can be transformed into: * ( ) π Sxxy ij = M E msxxy ij = M E lim mxi mxj myk m m where m indicates the considered value that belongs to m th section. E means ensemble average and is defined by: M [ ] (4) E A A (5) M M m M m= m [ ] [ ] Similarly, we write: Sxxy ij = M E msxxy ij π k * ( ) * = ME lim ωmxi mxj Hnp mxn mxp m m (6) where i, j,k,n,p all are discretized frequencies and these frequencies have the following relations: k = i j, n = k+ p Eq. leads to: ( ) Sxxy ij Hij = (7) Sxx isxx j he contents of the square brackets in Eq. 6 consist of self-terms and cross-terms. In the remainder we shall use a Newman s approximation for the estimation of the coefficients of the quadratic transfer function thereby limiting the total amount of unknowns. In the case that it is appropriate to apply the Newman s approximation, an equation ( ) Hij P l, l ( i+ j) (8) can hold, where P l is real and Q l is imaginary and zero. Eq. 7 gives the following formula to estimate P l : Re[S xxy ij] Pl = (9) S S xx i xx j his expression still includes the cross-terms mentioned in the previous section and that makes the accuracy of estimation of QF questionable. But due to the Newman s approximation, we can exclude the cross terms as follows. First of all, Eq. 6 can be rewritten as follows: Sxxy ij = M E msxxy ij k = i j π k * * = ME lim ω mxi mxj Pl mxn mxp n = k+ p m m ( n p )/ l = + π k * * = M E P msxx i msxx j R lim m Xi m Xj m Xn mx l + l ω p m m n i,p j () where R l is a real constant. Similarly, S = ME ms xxx ij xxx ij k = i j π k * * = ME lim ω mxi mxj mxn mxp n = k+ p () m m ( n p )/ l = + π k * * = M E msxx i msxx j lim m Xi m Xj m Xn m X + ω p m m n i,p j Paper No. 6-JSC-435 Bunnik

From this equation, we obtain: π k * * lim ω m Xi m X j mxn m Xp = ms xxx ij msxx i msxx j m m n i,p j Substituting Eq. into Eq., we have: () estimated drift forces. We used a main diagonal of P-matrix shown in Fig. to generate the force numerically. S = E S xxy ij M m xxy ij { xxx ij } = EP S S + R S S S M l m xx i m xx j l m m xx i m xx j he above equations lead to: (3) M ERe[ ms xxy ij] P l = + M E msxx i msxx j M m M E Im[ E Re[ S ] ms xxy ij]. (4) xxx ij ME Im[ ms M E msxx i ms xx j xxx ij ] he first term of the right hand side of this equation is the same as Eq. 9. he nd term indicates the effect of the cross terms. Note that Eq. 4 just depends on l because of the Newman s approximation. In the case that the output signal is purely the squared input signal, Eq. 4 gives us P = l. At the frequencies where MEIm[ ms xxx ij ] =, Eq. 4 cannot hold. In that case, we just use the st term of the right hand side of Eq. 4, which is basically in the frequency range where the group energy is zero. In the practical analysis, we used not only the ensemble averaging but also the spectral window to obtain a smooth P l. In the right hand side of Eq. 4 we just simply applied the Hanning spectral window. Example of the Estimation Procedure Here we show the examples of the estimation. Waves and forces are generated numerically in which we used the Newman s approximation to generate the force from the wave. In the examples, we employed irregular waves with JONSWAP type and BOX type white noise spectrum as shown in Fig.. Fig. : Main diagonal of P-matrix his P matrix corresponds to a barge-shaped FPSO in deep water. he following random waves and variations were used for the simulation. A 3-hour wave realization from a survival wave spectrum (Hs = m, p =.3 s) A 7-hour wave realization from a survival wave spectrum (Hs = m, p =.3 s) A3 3-hour wave realization from a white noise wave spectrum (Hs = m with energy between.4 and. rad/s) A4 7-hour wave realization from a white noise wave spectrum (Hs = m with energy between.4 and. rad/s) A: 3-hour survival wave case Fig. 3 shows the comparison of the estimated and the theoretical main diagonal of the P-matrix. he estimated values were derived using 3- hours survival wave. In the estimation, we divided the time traces into 6 sections ( M = 6). After the ensemble averaging, we applied Hanning spectral window 3 times (NH = 3). wave spectral density [m s] 6 5 4 3 white noise survival wave.5.5 Fig. : heoretical wave spectrums he wave is generated with the time step of.5[s] and total length of time trace was 3 [hours] and 7 [hours]. he 3-hour length was chosen because this is a typical test length used in ocean basins. he 7-hour length was chosen to see the effect of a longer duration on the Fig. 3: Comparison of theoretical and estimated P in 3-hour survival case. he estimated P-matrix was used to calculate the wave drift force. In Fig. 4, we compared wave drift force time trace that were computed with theoretical and with estimated P-matrix. he solid line means the wave drift force generated with the theoretical P-matrix and the dashed line indicates the computed drift force with the estimated P-matrix. hese two lines show a rather good agreement. Paper No. 6-JSC-435 Bunnik 3

A3: 3-hour white noise case Fig. 7 and Fig. 8 are for 3-hours white noise case. Here we estimated with M = 6 and NH = 3. In Fig. 7, we observe a considerable difference between the theoretical and the estimated transfer functions. he wave drift force time trace shows a reasonable agreement. Fig. 4: Comparison of time trace of wave drift forces generated with theoretical and with estimated P in 3-hour survival case. A: 7-hour survival case Here the estimated P was obtained using 7-hour survival wave (M = 8, NH = 3). Fig. 5 shows a good agreement between the theoretical and the estimated P-matrix. his figure also shows that we can estimate the P-matrix more accurately and in the wider range in 7-hour case than in the 3-hour case Fig. 6 shows the time traces of the wave drift forces computed with theoretical and the estimated P-matrix. he wave drift force time trace shows very good agreement. Fig. 7: Comparison of theoretical and estimated P in 3-hour white noise case. Fig. 8: Comparison of time trace of wave drift forces generated with theoretical P and with estimated P in 3-hour white noise case. A4: 7-hour white noise case Fig. 5: Comparison of theoretical and estimated P in 7-hour survival case. Fig. 9 shows a comparison between the theoretical and the estimated transfer function in 7-hour white noise case (M = 8, NH = 3). Fig. shows the time traces of the wave drift forces estimated with theoretical and the estimated P-matrix. In Fig. 9, we observe a good agreement between the exact and estimated P values. he wave drift force time trace shows a very good agreement. Fig. 6: Comparison of time trace of wave drift forces generated with theoretical P-matrix and with estimated P-matrix in 7-hour survival case. Fig. 9: Comparison of theoretical and estimated P in 7-hour white noise case. Paper No. 6-JSC-435 Bunnik 4

he wave drift forces in an irregular sea can be described by the following formula: N N () t = ζ iζ jpij ( ωi ωj ) t + εi ε j ) + ζiζ jqij sin( ωi ωj ) t + εi ε j ) F i= j= cos (7) Fig. : Comparison of time trace of wave drift forces generated with theoretical P and with estimated P in 7-hour white noise case. Method Estimating wave drift force transfer function by direct simulation In this section, an approach is described with which the wave drift force transfer functions can be estimated from the measured motion response and the wave elevation. he method also offers the possibility to estimate the total mass and linear and quadratic damping. A short description of the method is given by the following steps:. he wave drift force is estimated from the measured motion, velocity and acceleration and the mooring stiffness, and a priori guess on the damping values and total mass.. One value in the P matrix is set to. he remaining values are set to zero. he resulting wave drift force time trace is computed. his is done for all entries and also for the Q matrix. he total wave drift force is a superposition of all these time traces, multiplied by the actual value of the transfer function at the specific P or Q matrix entry. 3. With a least square method, appropriate multiplication coefficients are determined such that the difference between the measured wave drift force and the computed wave drift force is minimized. Estimation of force on vessel he wave forces on the vessel can be estimated from the measured motion, using the following equation of motion (surge is assumed): () () F = Mx & + b x + b x x& + cx (5) Where: F x = wave force [kn] M = total mass (including added mass) [tons] b () = linear damping coefficient [kns/m] b () = quadratic damping coefficient [kns /m ] c = linear mooring stiffness [kn/m] x = surge motion [m] Prior to the analysis, the measured motions are low-pass filtered such that the motion is only caused by the low-frequency wave drift forces. he motion part due to the wave frequency forces is filtered out. Alternatively, the forces excluding the inertia force and damping force can be computed. his way, the total mass and the damping coefficients can be estimated as well, together with the wave drift force transfer function. In this paper, only results are shown based on theoretical wave drift force time traces. Computation of wave drift force An irregular sea can be described by a sum of harmonic components: N i= ( ) ζ = ζ cos ω t +ε,withζ = S( ω ) δω (6) i i i i i he transfer function is described as a real matrix, the P matrix and an imaginary one called the Q matrix. he P matrix corresponds to the part of the wave drift force in phase with the wave groups, the Q matrix with the part out of phase with the wave groups. Using the Discrete Fourier ransform the number of frequency components is equal to the number of time steps. Instead of defining the transfer function at each discrete frequency ω i and ω j, the transfer functions are defined on a coarser frequency mesh (typically with a spacing of.5 to.5 rads - ), depending on the natural period of the system. he transfer function at in-between frequencies is then found by a D linear interpolation. he present method estimates the P and Q matrix on the coarse mesh. he advantage of the followed approach is that the transfer function is only determined at the relevant resolution and not subject to a resolution prescribed by the direct FF. Due to the limited number of coefficients that need to be estimated, a strict application of a Newman's approximation was not necessary. herefore, all the coefficients of the real and imaginary quadratic transfer function could be estimated. Least squares method By setting one value in the P matrix to one and the others to zero and computing the resulting wave drift force, the influence of one entry in the P matrix to the wave drift force can be found. By doing this for all relevant values in the P matrix (the ones where there is wave group energy present), all the individual influences can be computed. he same can be done for the Q matrix. All these influences are now multiplied by actual P and Q values. he P and Q values are chosen such (using a least squares method) that the difference between the measured wave drift force and the computed wave drift force is minimized in a least squares sense. he method explicitly uses the fact that the P matrix is symmetrical, and the Q matrix asymmetrical. Results In order to test the method several wave drift force time traces were generated. he transfer functions used to compute these time traces were obtained from a diffraction analysis for a loaded barge shaped FPSO in deep water (head seas) as indicated in Figure. As done for the Cross-bi-spectral method, the following random wave realizations were generated B 3-hour wave realization from a survival wave spectrum (Hs = m, p =.3 s) B 7-hour wave realization from a survival wave spectrum (Hs = m, p =.3 s) B3 3-hour wave realization from a white noise wave spectrum (Hs = m with energy between.4 and. rad/s) B4 7-hour wave realization from a white noise wave spectrum (Hs = m with energy between.4 and. rad/s) he wave spectra and realizations of the wave and the wave drift forces were exactly the same as in the cross-bi-spectral method, enabling a direct comparison between the two methods. hese wave realizations were subsequently used, together with the theoretical transfer function, to compute the wave drift force time traces. Newman s approach was used. he wave drift forces were computed up to a difference frequency of. radians per second. Paper No. 6-JSC-435 Bunnik 5

he wave elevation and wave drift force time traces were subsequently used to estimate the quadratic transfer function using the method described before. In the estimate a difference frequency of.55 radians per second was used. he theoretical wave drift force time traces contained energy up to. rad/s. herefore, a total of diagonals of the P matrix were estimated (difference frequencies,.5. rad/s). B: 3-hour wave realization from a survival wave spectrum (Hs = m, p =.3 s) Fig. shows the first diagonals of the P matrix (estimated versus theory) using the 3-hour survival wave: 5-5 - -5 - -5-3 -35 difference frequency rad/s -4..4.6.8 difference frequency.5 rad/s 5-5 - -5 - -5-3 -35-4..4.6.8 Fig.. Estimated Quadratic Wave Drift Force ransfer Function he estimated P matrix was then used (with the wave elevation time trace) to compute the wave drift force time trace. Fig. shows the theoretical wave drift force and the wave drift force according to the estimated P matrix (only first hour): w a v e d rift fo rc e [k N ] - -4-6 -8 - theory computed with estimated P - 5 5 5 3 35 time [s] Fig. Wave Drift Force ime Series from Estimated QF hese Figures show that: - A good estimate of the transfer function is obtained in the frequency region where most of the energy is located. In the frequency region with a small amount of energy (tail of the spectrum) there is a considerable scatter in the results, especially on the main diagonal of the P matrix. - he wave drift force time traces show a good agreement. B: 7-hour wave realization from a survival wave spectrum (Hs = m, p =.3 s) Fig. 3 shows the first diagonals of the P matrix (estimated versus theory) using the 7-hour survival wave: 5-5 - -5 - -5 difference frequency rad/s -3..4.6.8 difference frequency.5 rad/s 5-5 - -5 - -5-3..4.6.8 Fig. 3 Estimated Quadratic Wave Drift Force ransfer Function Fig. 4 shows the theoretical wave drift force and the wave drift force according to the estimated P matrix (only the first hour is shown): w a v e d rift fo rc e [k N ] -.5 - -.5 x 4 - theory computed with estimated P 5 5 5 3 35 time [s] Fig. 4 Wave Drift Force ime Series from Estimated QF Paper No. 6-JSC-435 Bunnik 6

hese Figures show that: - here is still a large scatter in the estimate of the main diagonal, but smaller than in the 3-hour case - he estimates of the other diagonals become better compared to the 3-hour case but differences can still be observed for the higher frequencies - he wave drift force time trace shows a very good agreement B3: 3-hour wave realization from a white noise wave spectrum hese Figures show that: - A very good agreement between the theoretical and estimated transfer functions is found - he wave drift force time trace shows a very good agreement B4: 7-hour wave realization from a white noise wave spectrum Fig. 7 shows the first diagonals of the P matrix (estimated versus theory) using the 7-hour white noise wave: he following Figure shows the first diagonals of the P matrix (estimated versus theory) using the 3-hour survival white noise wave: 5-5 difference frequency rad/s 5-5 - -5 difference frequency rad/s - -5 - -5 - -5-3..4.6.8 difference frequency.5 rad/s 5-5 -3..4.6.8 difference frequency.5 rad/s 5-5 - -5 - -5 - -5 - -5-3..4.6.8 Fig. 5 Estimated Quadratic Wave Drift Force ransfer Function he estimated P matrix was then used (with the wave elevation time trace) to compute the wave drift force time trace. he following Figure shows the theoretical wave drift force and the wave drift force according to the estimated P matrix: w a v e d rift fo rc e [k N ] - -4-6 -8 theory computed with estimated P - 5 5 5 3 35 time [s] Fig. 6 Wave Drift Force ime Series from Estimated QF -3..4.6.8 Fig. 7 Estimated Quadratic Wave Drift Force ransfer Function - he scatter becomes very small and an excellent agreement is obtained over the complete frequency range of the white noise spectrum. Fig. 8 shows the theoretical wave drift force and the wave drift force according to the estimated P matrix (only the first hour is shown): w a v e d rift fo rc e [k N ] - -4-6 -8 - - theory computed with estimated P -4 5 5 5 3 35 time [s] Fig. 8 Wave Drift Force ime Series from Estimated QF Paper No. 6-JSC-435 Bunnik 7

DISCUSSION AND CONCLUSIONS wo methods of estimating the wave drift force quadratic transfer function have been presented. Both the estimation procedure based on a cross-bi-spectral analysis techniques and the estimation procedure based on direct simulation can give larger scatter in the QF coefficients on the diagonal. his behaviour is known since in order to estimate the diagonal (i.e. the mean) of the QF as a limiting case of the group frequency going to zero. his leads to coefficients with a limited number of statistical relevance. he agreement already becomes much better once the difference frequency is larger than.5 rad/sec. In case of a white noise excitation the resulting estimates of the coefficients become much more scattered using the cross-bi-spectral analysis. It seems therefore that applying wave spectra give much more band limited effects than a white noise excitation, which covers a larger range of group frequencies. However, the use of a white noise spectrum gives much better results compared to using a survival spectrum when the direct simulation is used. his is most likely related to the conditioning of the least squares matrix in this method, but this needs to be investigated further. Also in both methods it becomes apparent that a substantial increase in the length of the time series leads to better estimates of the QF. he rms of the resulting coefficients for both methods is in this case the same. his is obvious since more statistical information is gathered (number of long-period oscillations in the time traces). In this paper it (at least in of the methods) it has been assumed that Newman s approximation can be applied. his is of course a non-valid approach in case of shallow water and for several types of floating structures (semi-submersibles). his method, therefore, needs to be extended further. REFERENCES Bendat, J. and L.Piersol (): Random Data: Analysis and Measurement procedures. Wiley Interscience 3 rd edition. Bendat, J. (99): Non Linear System Analysis and Identification from Random Data. Wiley Interscience USA. Brillinger D.R. (98): ime Series: Data Analysis and heory SF Holden USA. Dalzell, J.F. (97): Application of cross-bi-spectral analysis to ship resistance in waves, Stevens Institute of echnology NJ, Report SI-DL-7-66 Dalzell, J.F. (975): he applicability of the functional polynomial input-output model to ship responses in waves. Stevens Institute of echnology NJ,SI-DL-75-794, Huijsmans, R.H.M. and J.A. Pinkster (99). he Wave Drift Forces on Ships in Shallow Water. Proceedings of the Boss Conference London. Kim, Y. and Powers (979) : Digital Bispectral Analysis and its application to non-linear wave interaction. IEEE rans. on Plasma vol. pp -3. Stansberg, C. (997) : Linear and non-linear System Identification in Pinkster, J.A. (98): he low frequency second order wave drift forces on ships. PhD hesis Delft Model esting. Proceedings of the int. Conf. on Non-linear Aspects of Physical Model ests. Stansberg, C.. (994): Low frequency excitation and damping characteristics of a moored semi-submersible in irregular seas. Estimation from model test data. Proceedings of the Boss 994 Conf. MI USA. ick,l.j. (96): he estimation of ransfer Functions of Quadratic Systems. echnometrics, 4(7) 746-765. Paper No. 6-JSC-435 Bunnik 8