## 3 Problem Statement
### Barge-Based FOWTs' Motions Analysis by Response Amplitude Operators  
Utilization of RAOs for the evaluation of the FOWTs' motion in various sea states is crucial. Researchers have used RAOs in different applications for motion evaluation of marine systems. For instance, [PERSON] et al. in [29] applied RAOs to evaluate a hybrid wind-wave energy converter and [PERSON] et al. in [30] obtained RAOs to analyze the dynamic response of OC3-Hywind spar buoy. In this section, the procedure of plotting the RAOs for different modes of the input-output system is explained. The procedure of obtaining RAOs in frequency-domain for translational and rotational states of the system is as follows. First, geometric data of the standard barge platform, open-OWCs-based barge platform and closed-OWCs-based barge platform has been employed in order to conduct advanced WAMIT calculations to present added-mass, damping, hydrostatics and hydrodynamic matrices. The obtained matrices have been introduced to nonlinear equations of motions with the aim of achieving system's outputs. RAOs of each mode of the system have been calculated by the auto-spectral density of the input (wave elevation) and the cross-spectral density of the input/output (system responses) with this equation [30]:  
\[RAO=\frac{S_{xy}(\omega)}{S_{xx}(\omega)} \tag{41}\]  
where \(S_{xy}(\omega)\) and \(S_{xx}(\omega)\) are the cross-spectral and auto-spectral densities of the input \(x(t)\) and the output \(y(t)\), in the frequency domain, respectively. \(S_{xy}(\omega)\) and \(S_{xx}(\omega)\) is expressed by:  
\[S_{xy}(\omega)=\frac{1}{M}\sum_{s=1}^{M}\gamma^{[s]}(r)\tilde{X}^{[s]}(r) \tag{42}\]  
\[S_{xx}(\omega)=\frac{1}{M}\sum_{s=1}^{M}X^{[s]}(r)\tilde{X}^{[s]}(r) \tag{43}\]  
here \(X^{[s]}\) is the Fast Fourier Transform (FFT) spectrum of segment \(s\). \(M\) is the number of simulations of the procedure and \(r\) is the random noise sequence. \(\tilde{X}\) is the complex conjugate of \(X\). In this research, the procedure has been done for six times (\(M=6\)). Also, white-noise wave inputs with six different seeds have been produced. As it can be seen in Figure 3, the wave input as white-noise is plotted for 8000 s. To ensure that the transient effects have not been considered, 2000 s were removed. Besides, the considered wave input amplitude is one meter to guarantee the linear wave theory. In the end, the average sets are calculated to smooth the spectrum.Also, this averaging the peaks and drops vanish, lets the histogram be concentrated nearby its mean value and decreases the leakage effects with a lower noise sensitivity [31]. Figure 3: Wave elevation as white noise. Heading waves direction of zero degree with still air is two environmental conditions that have been taken into account. Besides, nonlinearities because of the flexible tower, hydrodynamic loads, viscous effect and mooring lines have been considered. In this work, RAOs of eight states of the system consisting of surge, sway, heave, side-to-side, roll, pitch, yaw and fore-aft are performed, shown in Figure 4a-h. The period range of RAOs are between 2 s and 35 s. In Figure 4, there have been shown three curves in green, red and blue dash representing for standard barge platform, open-OWCs-based barge platform and closed-OWCs-based barge platform, respectively. An important point realized from the figure, as could be expected, is that the RAOs' curves for the standard barge platform and closed-OWCs-based barge platform are almost identical. It means that the behaviour of the standard barge and closed-OWCs-based barge platforms are the same in different modes. Figure 4a shows the surge RAO. It is noticeable that the surge RAOs in three platforms for period ranges of 2-7.5 s and 15-35 s are the same. It is interpreted that all platforms have the same behaviour in surge for those period ranges. There are the surge resonance frequencies at 11.9 s, 12.12 s and 12.8 s for standard barge, closed-OWCs-based barge and open-OWCs-based barge platforms, respectively.