Semisubmerged Ships 547 
(z, = A) parallel to the undisturbed free surface. For the heave and pitch analysis assume 
that there are no lateral excursions. 
Stated briefly, the equations of motion for heave and pitch are 
m'(w' - q') = Z'(w',q',w',q',z,,0) (2) 
Lig! SE wiya’ omg", 2150) (3) 
where m' is the nondimensional mass of the craft (2m/e°), w’ is the vertical velocity of 
the center of gravity in fraction of speed (U, positive downward), q' is the nondimensional 
angular velocity about the transverse (y) axis, zi is the instantaneous distance from the 
free surface to the center of gravity in fraction of the length of the craft (2, positive down- 
ward), @ is the pitch angle (positive nose-up), Ti is the nondimensional mass moment of 
inertia about the y axis (I, =] y/(e/2)45), Z' is the nondimensional total vertical force 
(force 1/2e¢7U?), M' is the noddimenaiaaal total pitching moment (about the y axis), and 
the dots above a symbol indicate differentiation with respect to nondimensional time 
(s = Ut/4). 
Motion near the free surface induces forces and moments that are dependent upon the 
distance (x,) and possibly on the pitch angle (0). These dependencies are, of course, not 
present when the ship is deeply submerged. 
When the ship is near the free surface, the force and moment become dependent on z,. 
There is also a possible dependence of the force and moment upon @, which does not arise 
in the case of deeply submerged bodies. In view of the dependence on depth below the sur- 
face, it is advantageous to write the equations with respect to an axis system (x 595925) 
that is fixed in the free surface, rather than with respect to a system (x,y,z) that is fixed in 
the body as is usually done. Expanding the right sides of Eqs. (2) and (3) in Taylor expan- 
sion and retaining only the linear terms yields: 
(Zi-m')%, + Zit) + Zt zi 4+ 216+ (214+ 2/04 (21+ 238 = 0 
MZ) + HZ, + My Zo + Ma-1,)0 + (My, + Me + CM + Mg)2 = 0 65) 
where all the coefficients are derivatives with respect to the indicated subscripts (except m 
and I,). To obtain Eqs. (4) and (5), the kinematic relation between the motion referred to 
axes fixed in the craft and those fixed in the water surface has been used; therefore, 
WS oe) O and G=1G. 
In principle, these equations can be solved analytically, but the results are somewhat un- 
wieldy. They can also be solved on an analog computer. In our study we have attempted to 
glean some understanding from the analytical approach and have also used an analog for 
more rapid exploration of the influence of fin area and control on the response of the body 
when disturbed as it cruises at 30, 40, and 60 knots at a depth of 1.25 diameters (to the cen- 
terline). The following discussion will first reveal what has been learned from the hand- 
turned mathematics and will be followed by a recounting of the results of the analog studies. 
