## 2 Ship Hydroelasticity Theory in Irregular Waves
### Solution of Hydroelastic Motion Equation  
The hydroelastic motion governing equation of ship advancing in waves can be expressed as follows:  
\[[\mathbf{m}]\ddot{[\mathbf{p}]}+[\mathbf{c}][\dot{\mathbf{p}}]+[\mathbf{k}] [\mathbf{p}]=[\mathbf{F}_{I}]+[\mathbf{F}_{S}]+[\mathbf{F}_{D}]+[\mathbf{F}_{ R}]+[\mathbf{F}_{SI}]+[\mathbf{F}_{GW}] \tag{6}\]  
where \([\mathbf{m}]\), \([\mathbf{c}]\) and \([\mathbf{k}]\) are the generalized structural mass, damping and stiffness matrices, respectively; \([\mathbf{p}]\) denotes the generalized principal coordinate column matrix; the fluid force terms \([\mathbf{F}_{I}]\), \([\mathbf{F}_{S}]\), \([\mathbf{F}_{D}]\), \([\mathbf{F}_{R}]\), \([\mathbf{F}_{SL}]\) and \([\mathbf{F}_{GW}]\) on the right-hand side of Equation (6) are Froude-Krylov force, hydrostatic restoring force, diffraction force, radiation force, slamming load and green water load column matrices, respectively. The structural dynamic part on the left-hand side of Equation (6) is solved based on the linear structural mechanics theory. The hydrodynamic part on the right-hand side of Equation (6) is solved by nonlinear fluid mechanics theory. The nonlinear effects are considered as a compromise between the calculation accuracy requirement and time cost. Therefore, the Froude-Krylov force, hydrostatic restoring force, slamming loads and green water loads are calculated on the instantaneous wetted surface of the ship to include the nonlinearities associated with hull geometry. In contrast, the diffraction force and radiation force, which are obtained by solving the free-surface Green function, are remain as linear and calculated on the mean wetted ship surface in calm water. The calculation of generalized fluid force is described as follows. (1) Hydrostatic restoring force  
The nonlinear hydrostatic restoring force acting on the elastic hull is calculated on the instantaneous wetted body surface:  
\[F_{S}^{r}(t)=-\rho g\sum\limits_{k=1}^{m}p_{ka}\iint\limits_{S(t)}\mathbf{n} \mathbf{u}_{r}w_{k}ds-\int\limits_{L}F_{g}(x)w_{r}(x)\mathrm{d}x(r=1,2,\cdots,m) \tag{7}\]  
where \(S(t)\) denotes the instantaneous wetted body surface of hull in waves, \(p_{ka}\) is the \(k\)-th order principal coordinate amplitude, \(w_{k}\) denotes the \(k\)-th order mode of vertical displacement of the hull, \(F_{g}\) denotes the sectional hull weight. (2) Incident wave force  
For ships sailing in irregular waves, the incident wave force is obtained by integrating the contribution of each component of regular wave with different wave frequency, amplitude and phase. The following equation is used to obtain the incident wave force induced by irregular waves:  
\[F_{I}^{r}(t)=-\rho\sum\limits_{i=1}^{M}\zeta_{ai}\iint\limits_{S(t)}\mathbf{ n}\cdot\mathbf{u}_{r}\bigg{(}\mathrm{i}\omega_{i}-U\frac{\partial}{\partial x} \bigg{)}\phi_{0i}\mathrm{d}s(r=1,2,\cdots,m) \tag{8}\]where \(\zeta_{ali}\), \(\omega_{i}\), \(\phi_{0i}\) are the wave amplitude, frequency and incident potential of _i_th sub-wave component, respectively. (3) Diffraction wave force  
Since the diffraction wave force is calculated on the static mean wetted surface, the time-domain convolution integral method is used to obtain the diffraction wave force induced by irregular waves:  
\[F_{D}^{r}(t)=\int_{0}^{t}h_{D}^{r}(t-\tau)\zeta(\tau)\mathrm{d}\tau(r=1,2, \cdots,m) \tag{9}\]  
\[h_{D}^{r}(t)=\frac{1}{\pi}\!\int_{0}^{\infty}H_{D}^{r}(i\omega)\mathrm{e}^{ \mathrm{i}\omega t}\mathrm{d}\omega(r=1,2,\cdots,m) \tag{10}\]  
where \(h_{D}^{r}(t)\) is the impulse response function (IRF) of diffraction wave force for _r_-th mode of motion, and \(H_{D}^{r}(i\omega)\) is the frequency-domain response function of diffraction wave force for _r_-th mode of motion in regular waves. (4) Radiation wave force  
For ships in irregular waves, the wave memory effects resulted from the persistence of radiation waves generated by the ship's motion and its scattering of incident irregular waves play an import role in the determination of radiation wave force.