Frequency Domain Analysis |
 
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Frequency Domain Analysis |
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The underlying principles of analysis in the frequency domain are essentially the same for both traditional de-coupled analyses and coupled analyses. These principles are described in Frequency Domain Analysis, which summarises the theory underpinning frequency domain analysis. In this section, the important differences with respect to coupled analyses (as opposed to decoupled analyses) are described.
One of the major differences between the coupled analysis procedure in the time and frequency domains concerns the treatment of the frequency dependent added mass and radiation damping terms. In frequency domain analyses, the system response is solved for at each component harmonic in the spectrum of excitation forces. Hence, when solving for the response at a particular frequency, the appropriate frequency-dependent added mass and radiation damping terms for the floating body are simply incorporated in the global mass and damping matrices at the appropriate locations. Thus the solution fully accounts for the frequency-dependent nature of the added mass and radiation damping terms. Note that at frequencies below the user-specified low frequency damping cut-off frequency, the floating body added mass and damping terms correspond to the user-specified low frequency damping values.
First order (wave-frequency) loads are applied to each floating body by means of force RAOs, while mean (static) current and wind loads are applied to each floating body in the static phases of the frequency domain analysis.
If appropriate, hydrodynamic coupling between floating bodies is included in the frequency domain analysis by incorporation of the relevant terms from the added mass and damping co-influence matrices in the global mass and damping matrices respectively.
An important consideration is the treatment of low frequency second-order loads in frequency domain irregular sea coupled analyses. As second-order loads are inherently non-linear, a linearisation technique must be employed before these loads can be incorporated in frequency domain dynamic analyses. The procedure employed by Flexcom is as follows. First, the wave spectrum is discretised in the usual manner (using either the Equal Area or Geometric Progression discretisation procedures, as discussed in Spectrum Discretisation). Next, Flexcom identifies each frequency pair in the discretised seastate, and from them computes all difference frequencies. Note that, in general, there will be a significantly larger number of difference frequencies present than there are component harmonics in the discretised wave spectrum; in a wave spectrum with n component harmonics, there will be ½ n (n-1) frequency pairs. Each difference frequency is then added to the list of frequencies at which the equations of motion of the coupled system are to be solved in the irregular sea analysis. The magnitude of the harmonic excitation at each difference frequency is computed from the appropriate QTF values and component wave amplitudes. These are the QTFs and component wave amplitudes corresponding to the two frequencies in the frequency pair from which the difference frequency is derived. The magnitude of the low-frequency harmonic excitation is multiplied by the user-specified wave drift force calibration coefficients and the resulting low frequency harmonic excitation load applied to each floating body at each difference frequency for which the equations of motion are being solved.
Due to the non-linear nature of the floating body viscous damping loads, it is necessary to linearise these loads before they can be included in frequency domain dynamic analyses. The linearisation technique employed for the floating body viscous damping loads is based on the proven techniques currently used in Flexcom to linearise the Morison’s Equation drag force. These techniques are described in detail in Frequency Domain Analysis.