# Research on a Support System for Automatic Ship Navigation in Fairway
## 3 Mathematical Model of Ship Motion
### The Effect of Wind  
In a study performed by [PERSON] [35], the effect of wind on merchant ships was carried out at different laboratories. Empirical formulas were derived in order to calculate the force and the moment on any merchant ship induced by wind. By applying these formulas, the coefficients are determined as follows:  
\[\begin{split}\mathrm{C_{X}}=\mathrm{A_{0}}\;+\;\mathrm{A_{1}}\frac {2\mathrm{A_{L}}}{\mathrm{L_{OA}^{2}}}\;+\;\mathrm{A_{2}}\frac{2\mathrm{A_{T}} }{\mathrm{B^{2}}}\;+\;\mathrm{A_{3}}\frac{\mathrm{L_{OA}}}{\mathrm{B}}\;+\; \mathrm{A_{4}}\frac{\mathrm{S}}{\mathrm{L_{OA}}}\;+\;\mathrm{A_{5}}\frac{ \mathrm{C}}{\mathrm{L_{OA}}}\;+\;\mathrm{A_{6}}\mathrm{M}\\ \mathrm{C_{Y}}\;=\;\mathrm{B_{0}}\;+\;\mathrm{B_{1}}\frac{2\mathrm{A_{ L}}}{\mathrm{L_{OA}^{2}}}\;+\;\mathrm{B_{2}}\frac{2\mathrm{A_{T}}}{\mathrm{B^{2}}} \;+\;\mathrm{B_{3}}\frac{\mathrm{L_{OA}}}{\mathrm{B}}\;+\;\mathrm{B_{4}}\frac{ \mathrm{S}}{\mathrm{L_{OA}}}\;+\;\mathrm{B_{5}}\frac{\mathrm{C}}{\mathrm{L_{OA }}}\;+\;\mathrm{B_{6}}\frac{\mathrm{A_{SS}}}{\mathrm{A_{L}}}\\ \mathrm{C_{N}}\;=\;\mathrm{C_{0}}\;+\;\mathrm{C_{1}}\frac{2\mathrm{A_ {L}}}{\mathrm{L_{OA}^{2}}}\;+\;\mathrm{C_{2}}\frac{2\mathrm{A_{T}}}{\mathrm{B^ {2}}}\;+\;\mathrm{C_{3}}\frac{\mathrm{L_{OA}}}{\mathrm{B}}\;+\;\mathrm{C_{4}} \frac{\mathrm{S}}{\mathrm{L_{OA}}}\;+\;\mathrm{C_{5}}\frac{\mathrm{C}}{\mathrm{ L_{OA}}},\end{split} \tag{9}\]  
where \(\mathrm{C_{X}}\) is the coefficient of the longitudinal component of the wind force; \(\mathrm{C_{Y}}\) is the coefficient of the lateral component of the wind force; and \(\mathrm{C_{N}}\) is the coefficient of the wind-induced yawing moment. (A\({}_{0}\)-\(\mathrm{A_{6}}\), B\({}_{0}\)-\(\mathrm{B_{6}}\), C\({}_{0}\)-\(\mathrm{C_{5}}\)) are [PERSON]'s coefficients in the above equations.  
In order to define the wind-induced forces and moments acting on the ship, the following equations are used:  
\[\begin{split}\mathrm{X_{W}}\;=\;\mathrm{C_{X}}\frac{1}{2}\rho \mathrm{V_{R}^{2}}\mathrm{A_{T}}\\ \mathrm{Y_{W}}\;=\;\mathrm{C_{Y}}\frac{1}{2}\rho\mathrm{V_{R}^{2}} \mathrm{A_{L}}\\ \mathrm{N_{W}}\;=\;\mathrm{C_{N}}\frac{1}{2}\rho\mathrm{V_{R}^{2}} \mathrm{A_{L}}\mathrm{L_{OA}},\end{split} \tag{10}\]  
where, \(\mathrm{X_{W}}\) is the longitudinal component of wind force; \(\mathrm{Y_{W}}\) is the lateral component of wind force; \(\mathrm{N_{W}}\) is the yawing moment; \(\mathrm{V_{R}}\) is the relative wind speed on the ship; and \(\mathrm{L_{OA}}\) is the overall length of ship.  
In this research, ship motion is assumed to be affected by wind alone. However, in reality there are a number of different disturbances at work, such as currents and waves. Because the ship is navigating in a fairway, and free board is significantly more than the ship's draft, the effects of current and waves can be ignored. In cases of tankers or bunker vessels, the effects of current and waves should be taken into account in the ship dynamic model. To determine the effects of current and waves on ship motion, one must calculate the forces and moments induced by these disturbances. They would then be placed into the right side of Equation (2).