3.2 Hydrodynamic Coefficients

　

(1) Lift and drag of hydrofoil

　

 The coefficients of lift and drag of hydrofoil were calculated by the expression summarized by Cane [6] as follows;

　

　

where αT is the attack angle from zero lift angle, λ is the aspect ratio and CD0 is the section drag coefficient.

　

 K means the effect of lift loss due to pressure relief as a function of foil depth. Wadlin et al. [7] defined as follows:

　

　

 where, h is the depth of submergence and c is the chord length of the hydrofoil. Ω means the effect of tilt of the lift vector due to wave. Vladimirov [8] expressed as follow:

　

　

 σ is Prandtl's finite-span biplane factor and the approximation is given by:

　

　

Masuyama [9, 10] showed that these expressions gave good results by through the towing tank tests.

　

(2) Drag of strut

　

 The strut has a symmetrical section. Then the coefficient of section drag is obtained as follow by Hoerner [11]:

　

　

 where, t is thickness of the strut. The coefficient of spray drag is expressed again by Hoerner.

　

　

3.3 Equations of Motion

　

 In order to focus on the rolling stability and control capability of the boat, we express the equations of motion using four simultaneous differential equations such as surge, sway, roll and yaw. Therefore the terms concerning with the velocity W and angular velocity Q are eliminated from the equations. The equations are expressed thoroughly as follows:

　

surge:

　

　

sway:

　

　

roll:

　

(Ixx + ΣmyiZi² + Σmziyi²)Φ - ΣmyiziV = ΣYizi + ΣZiyi - mcgzcsinφc　(14)

　

yaw:

　

　

Φ and Ψ are Euler's angles and related as follows:

Φ = P

Ψ = R cosΦ　(16)

　

 In the equations (12) to (15), xi, yi and zi, are the coordinates of the acting point of hydrodynamic forces as above mentioned, and we assumed this point is on the center of the wetted surface area of hydrofoil or strut. For the main hydrofoil, in order to consider the effect of both rolling and yawing motions of the boat, we calculated considering the foil separately in half and the acting point of each part is on the 20% point from the tip. Xi, Yi and Zi are the hydrodynamic forces obtained from equation (4). m is mass of the boat and mc mass of the crew. φc is incline angle of the crew and we assumed that 70% of mc moves in the cockpit with his inclination, i.e., mc' = 0.7mc. xc and zc are the coordinates of representative point of moving mass of the crew, where zc, is negative. Figure 7 shows the definition of crew incline angle. myi, and mzi are added masses acting on the struts and foils in the direction of y axis and z axis, respectively. These added masses are evaluated assuming that the foil or strut blade as an elliptic plane.