The foil area Fy, contributing to the damping, is of course proportional to the foil area F: 
Ni: VA7F , (ny 
It was already stated that / :: ml? and PR :: mgl, so 
ae eae VE 
a Vml2mél mV 6)” (12) 
in es 
Combining Eqs. (5) and (6) with (12) leads to 
kg 
Jee eR 
Similarly it can be found that in case of heave 
ka 
d, = C| Fr. (14) 
Consequently the damping ratio is inversely proportional to the lift coefficient and the 
Froude number for both types of motion. 
This result might seem unexpected, the damping being proportional to speed, while the 
damping ratio is inversely proportional to the Froude number. The explanation is simple: 
an increase of speed of 10 percent corresponds with a decrease of submerged foil area of 
20 percent, if the lift coefficient remains constant (see Eq. (6)). For similar hydrofoil craft 
of differing size, 4,,,, kay» kid a and kq, are constant values. They may vary, however, 
for different hydrofoil systems. 
All elements necessary to calculate the motions of a hydrofoil craft are now available 
if the craft may be considered as a damped linearized spring and mass system. The dynamic 
amplitudes are then determined by the equations 
1 
Wg Z (1- w/ wat)? + 4d, W/ Wap (15) 
1 
z (16) 
Akay = V(1- w/w2,)7+ 4d,?w/ wz, § 
These equations have been plotted in Fig. 4 for various damping ratios. 
It is now possible to determine the vertical accelerations at a certain point of the craft. 
This point has been chosen near the bow at a distance from the center of gravity of 1/3, no 
passengers or crew being expected to be carried at a more forward point. The resulting 
accelerations have been calculated by adding vectorially the vertical acceleration caused 
by a variation of the pitch w and the vertical acceleration caused by a variation of heave 
Z. This is based on the assumption that pitch and heave are entirely independent of each 
other, which, although not correct, gives very acceptable results, as has been shown by 
Abkowitz [9]. 
