## 4 Results and Analysis
### Application to Planing Boats
Applying the correction due to aft suction according to [PERSON] [21]  
\[\mathrm{f}_{\mathrm{ys}}\ =\ \mathrm{C}_{\mathrm{tr}}(\mathrm{x})\mathrm{f}_{ \mathrm{y}} \tag{95}\]  
\[\mathrm{f}_{\mathrm{zs}}\ =\ \mathrm{C}_{\mathrm{tr}}(\mathrm{x})\mathrm{f}_{ \mathrm{z}} \tag{96}\]  
\[\mathrm{m}_{\mathrm{xs}}\ =\ \mathrm{C}_{\mathrm{tr}}(\mathrm{x})\mathrm{f}_{ \mathrm{z}} \tag{97}\]  
where \(\mathrm{f}_{\mathrm{ys}}\), \(\mathrm{f}_{\mathrm{zs}}\), and \(\mathrm{m}_{\mathrm{xs}}\) are the seccional forces of the boat, and \(\mathrm{C}_{\mathrm{tr}}\) is the correction due to aft suction according to [PERSON] [21]:  
\[c_{tr}(x)\ =\ tanh\bigg{(}\frac{2.5}{0.34B\ C_{p}}\mathrm{\chi}_{b}\bigg{)} \tag{98}\]  
Figure 18: Steady condition boat. where B is the beam of the boat, Cv is the velocity coefficient, and xb is the position of the section with reference to the interception of the aft and keel. The forces over the boat can be computed as:  
\[\text{F}_{\text{x}} = -\frac{1}{2}\rho\text{C}_{\text{f}}\text{A}_{\text{w}}\text{U}^{2} \tag{99}\] \[\mathrm{w}_{2}\ =\ \mathrm{w}^{\prime}-\frac{1}{2}\mathrm{v}^{\prime}\sin(2 \mathrm{\beta}_{2})+\mathrm{v}_{\mathrm{r}}\sin(\mathrm{\alpha})\cos(\mathrm{ \beta}_{2}) \tag{92}\]0 (100) \[\mathrm{w}_{2}\ =\ \mathrm{w}^{\prime}-\frac{1}{2}\mathrm{v}^{\prime}\sin(2 \mathrm{\beta}_{2})+\mathrm{v}_{\mathrm{r}}\sin(\mathrm{\alpha})\cos(\mathrm{ \beta}_{2}) \tag{92}\]1 (101) \[\mathrm{w}_{2}\ =\ \mathrm{w}^{\prime}-\frac{1}{2}\mathrm{v}^{\prime}\sin(2 \mathrm{\beta}_{2})+\mathrm{v}_{\mathrm{r}}\sin(\mathrm{\alpha})\cos(\mathrm{ \beta}_{2}) \tag{92}\]2 (102) \[\mathrm{w}_{2}\ =\ \mathrm{w}^{\prime}-\frac{1}{2}\mathrm{v}^{\prime}\sin(2 \mathrm{\beta}_{2})+\mathrm{v}_{\mathrm{r}}\sin(\mathrm{\alpha})\cos(\mathrm{ \beta}_{2}) \tag{92}\]3 (103) \[\mathrm{w}_{2}\ =\ \mathrm{w}^{\prime}-\frac{1}{2}\mathrm{v}^{\prime}\sin(2 \mathrm{\beta}_{2})+\mathrm{v}_{\mathrm{r}}\sin(\mathrm{\alpha})\cos(\mathrm{ \beta}_{2}) \tag{92}\]4  
where Fx is the force due to viscous drag, Fy is the normal force to the keel and parallel to the water plane, Fz is the normal force to the keel over vertical plane, Mx is the roll moment in the keel, My is the trim moment, Mz is the yaw moment evaluated at the interception between the keel and the aft, Cf is the frictional coefficient according to ITTC 1957 [27], Lk is the keel wetted length, and Aw is the wetted area of the hull. The forces in the body fixed coordinate system can be computed as:  
\[\left[ \begin{array}{c}\text{F}_{\text{xb}}\\ \text{F}_{\text{yb}}\\ \text{F}_{\text{zb}}\end{array} \right]=\left[ \begin{array}{ccc}1&0&0\\ 0&\cos(\phi)&-\sin(\phi)\\ 0&\sin(\phi)&\cos(\phi)\end{array} \right]\left[ \begin{array}{c}\text{F}_{ \text{x}}\\ \text{F}_{\text{y}}\\ \text{F}_{\text{z}}\end{array} \right]\left[ \begin{array}{c}\text{M}_{\text{ xb}}\\ \text{M}_{\text{yb}}\\ \text{M}_{\text{zb}}\end{array} \right]=\left[ \begin{array}{ccc}1&0&0\\ 0&\cos(\phi)&-\sin(\phi)\\ 0&\sin(\phi)&\cos(\phi)\end{array} \right]\left[ \begin{array}{c}\text{M}_{ \text{x}}\\ \text{M}_{\text{y}}\\ \text{M}_{\text{z}}\end{array} \right] \tag{105}\]  
where Fxb, Fyb, Fzb, Mxb, Mxb, Mxb, and Mxb are the forces in the in the body fixed coordinate system at the interception of the aft and keel. The non-dimensional force coefficients are defined as:  
\[C_{FXb} = \frac{F_{xb}}{\frac{1}{2}\rho B^{2}\text{U}^{2}} \tag{106}\] \[C_{FYb} = \frac{F_{yb}}{\frac{1}{2}\rho B^{2}\text{U}^{2}}\] (107) \[C_{FZb} = \frac{F_{zb}}{\frac{1}{2}\rho B^{2}\text{U}^{2}}\] (108) \[C_{MXb} = \frac{M_{xb}}{\frac{1}{2}\rho B^{3}\text{U}^{2}}\] (109) \[C_{MYb} = \frac{M_{yb}}{\frac{1}{2}\rho B^{3}\text{U}^{2}}\] (110) \[C_{MZb} = \frac{M_{zb}}{\frac{1}{2}\rho B^{3}\text{U}^{2}} \tag{111}\]  
where \(C_{FXb}\), \(C_{FYb}\), and \(C_{FZb}\) are the non-dimensional forces coefficients in x, y, and z in the body fixed coordinate system. \(C_{MXb}\), \(C_{MXb}\), \(C_{MYb}\), and \(C_{MZb}\) are the non-dimensional moment coefficients in x, y, and z in the body fixed coordinate system.