## 4 Results and Analysis
### Application to Planing Boats  
Figure 17 shows the coordinate system used in the study of planing hulls. The xyz system has its origin at the interception of the keel and aft, and it is fixed to the earth. The x axis is aligned with the forward velocity. The system \(x_{b}y_{b}z_{b}\) is fixed with the boat (BFCS) and rotates with it. The axis \(x_{b}\) is aligned with the keel, while the axis \(y_{b}\) is perpendicular to the symmetry plane of the boat and has its origin at the interception of the keel and aft. The system \(x_{cg}y_{cg}z_{cg}\) is fixed to the boat and has its origin in the CG of the boat. The axis \(x_{cg}\) is parallel to the keel. Figure 18 shows a planing boat sailing in steady condition, the boat has a total velocity U, trim angle \(\theta\), heel angle \(\phi\) and drift angle \(\psi\), \(L_{cg}\) and \(V_{cg}\) are longitudinal and vertical position respect to the system \(x_{b}y_{b}z_{b}\), and L is the length of the boat. Figure 16: Impact with three degrees of freedom, m = 293 kg, h = 0.61 m, \(\beta\) = 20\({}^{\circ}\), \(\phi_{0}\) = 5\({}^{\circ}\), \(Vcg\) = 0.216 m, and \(I_{xx}\) = 10.95 kg.m\({}^{2}\). (**a**) Horizontal acceleration during the impact. (**b**) Vertical acceleration during the impact. (**c**) Angular acceleration during the impact. (**d**) Horizontal velocity during the impact. (**e**) Vertical velocity during the impact. (**f**) Angular velocity during the impact. Figure 17: Boat coordinate systems. The results for two-dimensional impact can be used to study the dynamics of a planing boat through the application of slender body theory. The pressure in the hull is computed as:  
\[\frac{\mathrm{P}}{\mathrm{\rho}}\ =\ -\frac{\mathrm{D}\mathrm{\varphi}}{\mathrm{D} \mathrm{t}}-\frac{1}{2}\bigg{(}\frac{\mathrm{d}\mathrm{\varphi}}{\mathrm{d} \mathrm{y}}\bigg{)}^{2} \tag{89}\]  
where \(\frac{\mathrm{D}\mathrm{\varphi}}{\mathrm{D}\mathrm{t}}\) is the material derivative of the potential flow function from Equation (29). The velocity for each section of the hull is evaluated as:  
\[ \begin{array}{rcl}\mathrm{w}^{\prime}&=&\mathrm{w}+\mathrm{usin}(\mathrm{ \theta})-\Big{(}\mathrm{x}-\mathrm{L}_{\mathrm{cg}}\Big{)}\dot{\mathrm{\theta}} \\ &\mathrm{v}^{\prime}&=&\mathrm{v}+\Big{(}\mathrm{x}-\mathrm{L}_{\mathrm{cg}} \Big{)}\dot{\mathrm{\psi}}\end{array}  \tag{90}\]  
where \(\mathrm{w}^{\prime}\) is the vertical velocity of the boat section, \(\mathrm{v}^{\prime}\) is the horizontal velocity of the boat section, \(\mathrm{u}\) is the surge velocity of the boat, \(\mathrm{\theta}\) is the trim angle, \(\dot{\mathrm{\theta}}\) is the pitch velocity, \(\dot{\mathrm{\psi}}\) is the yaw velocity, and \(\mathrm{L}_{\mathrm{cg}}\) is the horizontal position of \(\mathrm{cg}\) from the interception of the keel and aft, including the roll velocity \(\mathrm{\omega}\):  
\[\mathrm{w}_{1}\ =\ \mathrm{w}^{\prime}+\frac{1}{2}\mathrm{v}^{\prime}\sin(2 \mathrm{\beta}_{1})-\mathrm{v}_{\mathrm{r}}\sin(\mathrm{\alpha})\cos(\mathrm{ \beta}_{1}) \tag{91}\]  
\[\mathrm{w}_{2}\ =\ \mathrm{w}^{\prime}-\frac{1}{2}\mathrm{v}^{\prime}\sin(2 \mathrm{\beta}_{2})+\mathrm{v}_{\mathrm{r}}\sin(\mathrm{\alpha})\cos(\mathrm{ \beta}_{2}) \tag{92}\]  
\[\mathrm{v}_{\mathrm{t}1}\ =\ -\mathrm{v}^{\prime}\cos(\mathrm{\beta}_{1})+ \mathrm{v}_{\mathrm{r}}\cos(\mathrm{\alpha}) \tag{93}\]  
\[\mathrm{v}_{\mathrm{t}2}\ =\ -\mathrm{v}^{\prime}\cos(\mathrm{\beta}_{1})- \mathrm{v}_{\mathrm{r}}\cos(\mathrm{\alpha}). \tag{94}\]  
The forces over the boat are computed by the integration of the sectional forces along the length.