## 2 Mathematical Model
### Equivalent Roll Damping Using the \(TGF\_N\_V\)  
The Bernoulli's equation with consideration of viscous dissipation effects of water can be written as  
\[\frac{\partial\psi}{\partial t}+\frac{P}{\rho}+\mathrm{g}\eta+\frac{1}{2}( \
abla\psi)^{2}+2\epsilon\omega_{0}\psi=0 \tag{17}\]  
where \(P\) is the overall pressure, which can be decomposed into the sum of hydrostatic pressure \(p_{0}\) and hydrodynamic pressure \(p\)  
\[P=p_{0}+p \tag{18}\]  
\[\frac{p_{0}}{\rho}+\mathrm{g}\eta=0 \tag{19}\]  
The hydrostatic pressure \(p_{0}\) relates only to the immersion depth of the wetted surface, while the roll damping associates only with roll angular velocity, according to Equation (1). Therefore \(p_{0}\) does not make contribution to the roll damping. Subtracting Equation (19) from Equation (17) yields  
\[\frac{\partial\psi}{\partial t}+\frac{p}{\rho}+\frac{1}{2}(\
abla\psi)^{2}+2 \epsilon\omega_{0}\psi=0 \tag{20}\]  
Then the roll moment of the floating body due to hydrodynamic pressure is  
\[f(t)=-\rho\int_{S_{\mathrm{B}}(t)}\frac{\partial\psi}{\partial t}n_{c}\mathrm{ d}s-\rho\int_{S_{\mathrm{B}}(t)}\biggl{(}\frac{1}{2}(\
abla\psi)^{2}+2 \epsilon\omega_{0}\psi\biggr{)}n_{c}\mathrm{d}s \tag{21}\]where \(n_{\rm c}\) is the generalized normal vector on the wetted surface defined as  
\[n_{\rm c}=\overset{\rightharpoonup}{r}\times\overset{\rightharpoonup}{n}=(x,y) \times\big{(}n_{x},n_{y}\big{)}=xn_{y}-yn_{x} \tag{22}\]  
To enhance the numerical stability, the first term on the right-hand side of Equation (21) can be transformed using [PERSON]'s transport theorem [25]  
\[\int_{S_{\rm B}(t)}\frac{\partial\psi}{\partial t}n_{\rm c}\mathrm{d}s=\frac{ \mathrm{d}}{\mathrm{d}t}\int_{S_{\rm B}(t)}\psi n_{\rm c}\mathrm{d}s-\int_{S_{ \rm B}(t)}\left(\overset{\rightharpoonup}{r}\times\
abla\psi\right)V_{n} \mathrm{d}s-\psi\dot{x}(t)(\overset{\rightharpoonup}{r}\times\overset{ \rightharpoonup}{j})\left| \begin{array}{c}B\\ A\end{array} \right. \tag{23}\]  
The calculation of \(\overset{\rightharpoonup}{r}\times\
abla\psi\) and \(|\
abla\psi|^{2}\) was not detailed in Duan [25]. In this paper we would like to provide a possible approach to calculate them. As shown in Figure 1, \(V_{\rm T}\) is the tangential velocity of water on the wetted surface, which can be obtained through the numerical difference  
\[V_{\rm T}^{i}=\frac{\psi_{i+1}-\psi_{i-1}}{2\Delta s} \tag{24}\]  
where \(\Delta s\) is the segment length, \(i\) is the segment index for the wetted surface. Now, one can obtain the following expressions:  
\[(\
abla\psi)^{2}={V_{n}}^{2}+{V_{\rm T}}^{2} \tag{25}\]  
\[\overset{\rightharpoonup}{r}\times\
abla\psi=(x,y)\times\big{(}\psi_{x},\psi_ {y}\big{)}=(x,y)\times\big{(}n_{x}V_{n}+n_{y}V_{\rm T},n_{y}V_{n}-n_{x}V_{\rm T }\big{)} \tag{26}\]  
where \(n_{\rm d}\) is the dot product of \(\overset{\rightharpoonup}{r}\) and \(\overset{\rightharpoonup}{n}\). Substituting Equations (23) into (21), and then substituting Equations (25) and (26) into the resulting equation, one gets roll moment  
\[ \begin{split} f(t)=-\rho\frac{\mathrm{d}}{\mathrm{d}t}\int_{S_{ \rm B}(t)}\psi n_{\rm c}\mathrm{d}s-\rho\int_{S_{\rm B}(t)}\left(\frac{1}{2} \big{(}{V_{\rm T}}^{2}-{V_{n}}^{2}\big{)}+2\epsilon\omega_{0}\psi\right)n_{ \rm c}\mathrm{d}s-\rho\int_{S_{\rm B}(t)}V_{n}V_{\rm T}n_{\rm d}\mathrm{d}s\\ +\rho\psi\dot{x}(t)x\left|\begin{array}{c}B\\ A\end{array}\right.\end{split}  \tag{27}\]  
Once the time series of the roll moment is obtained, the equivalent roll damping can be calculated using the cosine Fourier transform [16]  
\[b_{44}=\frac{1}{\pi}\int_{nT}^{(n+1)T}\mathrm{Im}\{f(t)\}\cos\omega t\mathrm{ d}t \tag{28}\]  
where \(T\) is the period of roll, \(\mathrm{Im}\{\cdot\}\) takes the imaginary part of the complex number. The integral in Equation (28) starts from the time \(nT\), when the time series has reached a steady state. The imaginary part of \(f(t)\) is selected for calculation due to that the imaginary part of roll (\(\mathrm{Im}\{\dot{\theta}\}\)) reaches to its maximum angular velocity at time \(nT\) and \((n+1)T\).