2.3.11. Response amplitude operators (RAOs)

2.3.11.1. Response amplitude operators (RAOs)

RAOs are transfer functions of the relation between the wave exciting forces and ship movements, used to determine the effect that a sea state will have upon the motion of a ship through the water. Calculation of Response Amplitude Operators (RAOs) in SeaFEM is done by analyzing the time series response of the ship, using a discretized white noise spectrum. This spectrum is defined by a number \(N^w=2^{m-1}\) of waves of equal amplitude and periods varying between the maximum and minimum values defined by the user. The values of these \(N^w\) periods are selected to match the discrete Fourier transform of the output signal, given by:

(2.660)\[\begin{aligned} & X = \sum_{n=0}^{N^w-1} x_n \cdot e^{-i2π∆f^* n} \end{aligned}\]

Given \(T_min\) and \(T_max\), the minimum and maximum periods of the analysis, the frequency increment is:

(2.661)\[\begin{aligned} & ∆f = \frac{f_{max} - f_{min}}{N^w-1} = \left( \frac{1}{T_{min}} - \frac{1}{T_{max}} \right) ⁄ \left( N^w - 1 \right) \end{aligned}\]

The well-known Fast Fourier Transform algorithms give a procedure to obtain an exact evaluation of the transfer functions defined above. This way, the time step and the total computing time are internally fixed to match the required sampling time and total sampling points. Then, the holding frequency \(∆f^*\) is evaluated as

(2.662)\[\begin{aligned} & ∆f^* = min(∆f ,f_{min}) \end{aligned}\]

And the discrete frequencies are:

(2.663)\[\begin{aligned} & f_n = n \cdot ∆f & n = 0,1,2,…,N^w-1 \end{aligned}\]

The required sampling frequency defines the time step as:

(2.664)\[\begin{aligned} & ∆t = \frac{1}{2∆f^* \cdot 2^m} \end{aligned}\]

The required number of sampling points defines the total calculation time as:

(2.665)\[\begin{aligned} & T = \frac{1}{∆f^*} \end{aligned}\]