222 Ø.W. Petersen et al. 0 0.2 0 0.5 1 1.5 2 x 10 11 F3 0 0.2 0 5 10 x 10 11 FR3 0 0.2 0 0.5 1 1.5 2 x 10 11 F2 Frequency [Hz] 0 0.2 0 1 2 3 x 10 10 FR2 Frequency [Hz] 0 0.2 0 2 4 6 x 10 9 F1 Spectral density [N2/Hz] 0 0.2 0 0.5 1 1.5 2 x 10 13 FR1 Spectral density [(Nm)2/Hz] a b Fig. 22.4 Diagonal elements of ƒpw.!/ for (a) the wave excitation forces (axial, lateral and vertical) and (b) wave excitation moments (torsion, vertical bending and lateral bending) at the middle pontoon The response of the structure to the simulated forces is obtained in the frequency domain. First, the generalized wave load pw;g.t/ 2 Rnm is defined as: pw;g.t/ Dˆ TSphpw.t/ (22.30) The load is then transformed into the frequency domain using the Fast Fourier Transform (FFT): pw;g.!/ DFFT pw;g.t/ (22.31) The modal response z.!/ is then obtained using the generalized transfer function Hg.!/: Hg.!/ D ! 2 I Cˆ T .Mh.!/ Mh0/ˆ Ci! Cˆ TCh.!/ˆ C 2 1 (22.32) z.!/ DHg.!/pw;g.!/ (22.33) It is emphasized that Hg.!/ is the force-to-motion transfer function and includes the hydrodynamic mass and damping [28], and is thus not the transfer function of the time-invariant system in Eq. (22.8). Rz.t/ is found using the inverse FFT: Rz.t/ DIFFT !2z.!/ (22.34) Finally, the modal acceleration output is transformed back to physical coordinates: y.t/ DSa Ru.t/ DSaˆRz.t/ (22.35) 22.3.3 Identification Model As mentioned, the model used for force identification can differ from the simulation model. A choice is made to only identify a subset of the 42 forces, namely the vertical and horizontal forces as well as the torsional moment, see Fig. 22.5. Thus np D 3 7 D21. The other force components are discarded mainly because their influence on the output is negligible and they are therefore considered too difficult to identify. From a practical standpoint, the neglected forces are also significantly smaller than those included in the identification. Nevertheless, by adding this error to the force terms in Eqs. (22.15) and (22.16), an additional modelling error is introduced. Also, the assumption that the measurement and modelling errors are uncorrelated is violated. The former effect can be accommodated for by means of the error covariance matrix Q, while there is no straightforward remedy to the violation of the correlation assumption. As will be shown later, though, the effect hereof seems negligible.
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