# Investigation on Ship Hydroelastic Vibrational Responses in Waves
## 2 Ship Hydroelasticity Theory in Regular and Irregular Waves
### Elasticity Theory in Ship Hydroelasticity
\begin{array}{c}B_{11}\text{sh}\left(\frac{ \lambda_{1}x}{l_{i}}\right),B_{12}\text{ch}\left(\frac{\lambda_{1}x}{l_{i}} \right),B_{13}\sin\left(\frac{\lambda_{2}x}{l_{i}}\right),B_{14}\cos\left( \frac{\lambda_{2}x}{l_{i}}\right)\\ B_{21}\text{ch}\left(\frac{\lambda_{1}x}{l_{i}}\right),B_{22}\text{sh}\left( \frac{\lambda_{1}x}{l_{i}}\right),B_{23}\cos\left(\frac{\lambda_{2}x}{l_{i}} \right),B_{24}\sin\left(\frac{\lambda_{2}x}{l_{i}}\right)\\ B_{31}\text{sh}\left(\frac{\lambda_{1}x}{l_{i}}\right),B_{32}\text{ch}\left( \frac{\lambda_{1}x}{l_{i}}\right),B_{33}\sin\left(\frac{\lambda_{2}x}{l_{i}} \right),B_{34}\cos\left(\frac{\lambda_{2}x}{l_{i}}\right)\\ A_{4}\end{array} \right]\left[ \begin{array}{c}A_{1}\\ A_{2}\\ A_{3}\\ A_{4}\end{array} \right] \tag{25}\]  
where:  
\[\left[ \begin{array}{c}B_{11},B_{12},B_{13},B_{14}\\ B_{21},B_{22},B_{23},B_{24}\\ B_{31},B_{32},B_{33},B_{34}\\ B_{41},B_{42},B_{43},B_{44}\end{array} \right]=\left[ \begin{array}{c}-\frac{l _{1}^{3}\lambda_{1}}{\beta^{4}E_{1i}},-\frac{l_{1}^{3}\lambda_{1}}{\beta^{4}E _{1i}},\frac{l_{1}^{3}\lambda_{2}}{\beta^{4}E_{1i}},-\frac{l_{1}^{3}\lambda_{2 }}{\beta^{4}E_{1i}}\\ \frac{l_{1}^{2}\left(\sigma+\lambda_{1}\right)^{2}}{\beta^{4}E_{1i}},\frac{l_{ 1}^{2}\left(\sigma+\lambda_{1}\right)^{2}}{\beta^{4}E_{1i}},\frac{l_{1}^{2} \left(\sigma-\lambda_{2}\right)^{2}}{\beta^{4}E_{1i}},\frac{l_{1}^{2}\left( \sigma-\lambda_{2}\right)^{2}}{\beta^{4}E_{1i}}\\ \frac{l_{1}\lambda_{1}\left(\sigma+\lambda_{1}\right)^{2}}{\beta^{4}},\frac{l_ {1}\lambda_{1}\left(\sigma+\lambda_{1}\right)^{2}}{\beta^{4}},-\frac{l_{1} \lambda_{2}\left(\sigma-\lambda_{2}\right)^{2}}{\beta^{4}},\frac{l_{1}\lambda_ {2}\left(\sigma-\lambda_{2}\right)^{2}}{\beta^{4}}\\ 1,1,1,1\end{array} \right] \tag{26}\]  
The components of matrix \([A]\) of the right hand side of Equation (25) are constants which can be solved by the boundary condition, and the relation between components of \([A]\) and \(C_{i}\) in Equation (23) can be given as follows:  
\[\left[ \begin{array}{c}C_{1}\\ C_{2}\\ C_{3}\\ C_{4}\end{array} \right]=\left[ \begin{array}{c}0,-\frac{l_{1}^{3}\lambda_{1}} {\beta^{4}E_{1i}},0,0\\ -\frac{l_{1}^{3}\lambda_{1}}{\beta^{4}E_{1i}},0,0,0\\ 0,0,0,-\frac{l_{1}^{3}\lambda_{2}}{\beta^{4}E_{1i}}\\ 0,0,\frac{l_{1}^{3}\lambda_{2}}{\beta^{4}E_{1i}},0\end{array} \right]\left[ \begin{array}{c}A_{1}\\ A_{2}\\ A_{3}\\ A_{4}\end{array} \right] \tag{27}\]  
Since these beam segments are linked together, we can adopt the Transfer Matrix Method to solve the overall vibrational characteristics of the whole Timoshenko beam of the hull. Note that for the oblique wave conditions, the twisting vibration modes are important and should be considered. Since this study concentrates on ship hydroelastic responses in head wave conditions, the 1D Timoshenko beam theory is adequate because only the vertical vibration mode exists. In addition, by adopting modal superposition formulation, the computational costs between above-mentioned two beam theories, namely the Euler-Bernoulli beam theory and the Timoshenko beam theory, are nearly the same, since the difference only lies in the generalized matrices \([\mathbf{m}]\), \([\mathbf{c}]\) and \([\mathbf{k}]\) in Equation (2), all of which can be conveniently obtained for both beam theories. This fact can also be considered as one advantage of adopting Timoshenko beam theory in the modal superposition method.