## 2 Ship Hydroelasticity Theory in Irregular Waves
### Solution of Hydroelastic Motion Equation
The radiation wave force in irregular waves can be expressed as:  
\[\mathbf{F_{R}}(t)=-[\mathbf{A}^{\infty}]\sqrt[\mathbf{B}^{\infty}]\sqrt[ \mathbf{B}^{\infty}]\sqrt[\mathbf{C}^{\infty}]\sqrt[\mathbf{p}(t)]\sqrt[ \mathbf{C}^{\infty}]\sqrt[\mathbf{p}(t)]\sqrt[\mathbf{p}(t)]\sqrt[\mathbf{p}(t -\tau)]\mathrm{d}\tau \tag{11}\]  
where the \([\mathbf{A}^{\infty}]\), \([\mathbf{B}^{\infty}]\) and \([\mathbf{C}^{\infty}]\) are infinite frequency added mass, damping coefficient and stiffness matrices caused by speed effects, \(\mathbf{K}(\tau)\) is the retardation function matrix and its component \(K_{rk}(\tau)\) can be calculated according to the Kramer-Kronig relation equation:  
\[K_{rk}(\tau)=\frac{2}{\pi}\!\int_{0}^{\infty}B_{rk}(\omega)\cos(\omega\tau)d\omega \tag{12}\]  
where \(B_{rk}(\omega)\) is the damping coefficient in frequency-domain. (5) Slamming loads  
In this study, the slamming loads for a ship sailing in irregular waves are obtained by momentum impact theory. The slamming force is determined by the momentum changing rate of the fluid around the bow region, which is expressed as follows:  
\[f_{slam}(x,t)=\frac{\mathrm{d}m_{\infty}(x)}{\mathrm{d}z}\left(\frac{ \mathrm{d}w_{rel}(x,t)}{\mathrm{d}t}\right)^{2} \tag{13}\]  
where \(m_{\infty}(x)\) is the sectional infinite frequency added mass, \(w_{rel}(x,t)\) is the real-time hull vertical displacement relative to the incident wave elevation and its time derivative can be expressed as:  
\[V_{z}(x,t)=\dot{w}_{rel}(x,t)=\sum_{r=1}^{m}w_{r}(x)p_{r}(t)-\zeta(x,t) \tag{14}\]  
where \(w_{r}(x)\) is the _r_th mode of ship vertical displacement or deformation. Then the generalized slamming force can be obtained by:  
\[F_{slam}^{r}(t)=\int_{L}f_{slam}(x,t)n_{var}(x)\mathrm{d}x \tag{15}\]  
where \(n_{var}(x)\) is _r_-th vertical component of coordinate of the unit normal vector \(\mathbf{n}\). (6) Green water loads  
The green water loads on deck are also calculated based on the momentum impact theory. The vertical impact force due to the presence of green water on deck is expressed as:  
\[F_{S}^{r}(t)=-\rho g\sum\limits_{k=1}^{m}p_{ka}\iint\limits_{S(t)}\mathbf{n} \mathbf{u}_{r}w_{k}ds-\int\limits_{L}F_{g}(x)w_{r}(x)\mathrm{d}x(r=1,2,\cdots,m) \tag{7}\]0  
where \(m_{gw}\) is the mass of the water on deck. Then the overall green water loads acting on the whole elastic hull can be obtained by integrating the 2D sectional force longitudinally in a similar manner as Equation (15). For the solution of the hydroelastic motion equation of Equation (6), the fourth-order Runge-Kutta algorithm is adopted in the present study. Then based on the modal superposition principle, the time-domain displacement \(w\)(\(x\),\(t\)), vertical bending moment \(M\)(\(x\),\(t\)), and shear force \(V\)(\(x\),\(t\)) along ship length can be written as follows:  
\[\left\{ \begin{array}{l}w(x,t)=\sum\limits_{r=1}^{m}p_{r}(t)w_{r}(x)\\ M(x,t)=\sum\limits_{r=1}^{m}p_{r}(t)M_{r}(x)\\ V(x,t)=\sum\limits_{r=1}^{m}p_{r}(t)V_{r}(x)\end{array} \right. \tag{17}\]  
where \(w_{r}(x)\), \(M_{r}(x)\) and \(V_{r}(x)\) are the \(r\)-th mode of vertical displacement, vertical bending moment and shear force, respectively.