# Investigation on Ship Hydroelastic Vibrational Responses in Waves
## 2 Ship Hydroelasticity Theory in Regular and Irregular Waves
### Elasticity Theory in Ship Hydroelasticity  
For simplicity, the whole ship structure can be treated as a long non-uniform beam, meaning that the beam theory can be used to describe the ship with the beam sectional parameters, such as the second moment of area, being obtained exactly based on the structural arrangement of the real ship section. This long ship hull beam as a whole can be mathematically described by the Euler-Bernoulli beam theory to predict global structural responses of the ship. However, either by adopting the usual finite element method or simplified linear modal superposition method, the whole ship hull beam needs to be further divided into beam segments, some of which may be short, and therefore, do not satisfactorily fulfill the requirement of adopting Euler-Bernoulli beam theory. Therefore, the present research treats the whole ship hull beam as a segmented Timoshenko beam which takes the effects of shear deformation and rotational inertia on the beam into consideration. The free vertical vibration of the \(i\)-th segment of the whole Timoshenko beam can be described by the following differential equation [30; 31]:  
\[EI_{i}\frac{\partial^{4}w_{i}}{\partial x^{4}}+\mu_{i}\frac{\partial^{2}w_{i}} {\partial l^{2}}-\left(\frac{EI_{i}\mu_{i}}{kGA_{i}}+J_{i}\right)\frac{ \partial^{4}w_{i}}{\partial x^{2}\partial l^{2}}+\frac{\mu_{i}J_{i}}{kGA_{i}} \frac{\partial^{4}w_{i}}{\partial t^{4}}=0 \tag{20}\]  
where \(w_{i}\) is the vertical displacement, \(E\) = 210 GPa and \(G\) = 80.77 GPa are respectively the elastic modulus and shear modulus which take the values of Q235 steel (since the typical structural components of the real ship across the chosen ship sections are made from Q235 steel), and are assumed to be invariant across the beam segments; \(I_{i}\) is the second moment of area which is considered to be constant along the \(i\)-th segment, \(\mu_{i}\) is the mass per unit length, \(J_{i}\) is the rotational inertia per unit length, \(A_{i}\) is the cross section area, and \(k\) is the Timoshenko shear coefficient and for the rectangular section \(k\) = 5/6. We assume that the solution of Equation (20) has the following form:  
\[w_{i}(x,t)=w_{i}(x)\sin(\omega_{0i}t+\varepsilon) \tag{21}\]  
where \(\omega_{0i}\) is the natural circular frequency of the vibration, \(\varepsilon\) is the initial phase angle. Substitute Equation (21) into Equation (20), we have:  
\[\frac{d^{4}w_{i}}{dx^{4}}+\frac{\omega_{0i}^{2}}{EI_{i}}\left(\frac{EI_{i}\mu _{i}}{kGA}+J_{i}\right)\frac{d^{2}w_{i}}{dx^{2}}-\frac{\mu_{i}\omega_{0i}^{2} }{EI_{i}}\left(1-\frac{J_{i}\omega_{0i}^{2}}{kGA}\right)w_{i}=0 \tag{22}\]  
The solution of Equation (22) is:  
\[w_{i}(x)=C_{1}\text{ch}(\lambda_{1}x/l_{i})+C_{2}\text{sh}(\lambda_{1}x/l_{i}) +C_{3}\cos(\lambda_{2}x/l_{i})+C_{4}\sin(\lambda_{2}x/l_{i}) \tag{23}\]  
where each \(C_{i}\) is a constant, \(\lambda_{i}\) and the coefficients can be expressed as:  
\[\left\{ \begin{array}{l}\lambda_{1,2}=\sqrt{\sqrt{\left(\beta^{4}+\frac{( \sigma-\tau)^{2}}{4}\right)\mp\frac{\sigma+\tau}{2}}}\\ \sigma=\frac{\mu_{i}\omega_{0i}^{2}}{kGA}l_{i}^{2}\\ \tau=\frac{l\omega_{0i}^{2}}{EI_{i}}l_{i}^{2}\\ \beta^{4}=\frac{\mu_{i}\omega_{0i}^{2}}{EI_{i}}l_{i}^{4}\end{array} \right. \tag{24}\]In addition to the vertical displacement mode \(w_{i}(x)\), the rotation angle mode \(\theta_{i}(x)\), vertical bending moment mode \(M_{i}(x)\) and shear force mode \(N_{i}(x)\) can be accordingly obtained, and the solution can be written in matrix form as follows:  
\[\left[ \begin{array}{c}w_{i}(x)\\ \theta_{i}(x)\\ M_{i}(x)\\ N_{i}(x)\end{array} \right]=\left[