Semisubmerged Ships 557 
making M_,,, change from positive to negative so that the second product in the above ine- 
quality contests with the magnitude of the positive product M,Z,,. Thus, it is possible to 
alter the stability in deep operation little beyond a certain maximum by a piling-on of fin 
area. At the same time, the quantity b increases enormously, and though b plays a minor 
role in determining stability in the deep operating case, in this case of coupled motion in 
the neighborhood of the free surface, it is very disadvantageous to make b large. This can 
be seen by again referring to Routh’s discriminant 
bed - (ad? + be) > 0 
which shows that the quadratic behavior of b can be overriding, provided the value of eis 
not too small. 
As we take area from the stern and put it on the bow, c decreases only about 20 percent, 
b is large and does not change, and e decreases by a factor of 8 (d changes but is not deci- 
sive). Thus, the role of M,, (which has, with equal areas forward and aft, a destabilizing 
tendency with regard to deep operation) is to provide a much softer effective spring value e 
to this fourth-order system and thus permit quick decay of oscillations. 
Control Effectiveness — The foregoing has shown that exponential instability will exist 
at Froude numbers below 0.3, that excessive tail area is not the answer at F = 0.6, and that 
reasonable tail area will only stabilize at F ~ 1.2 (which corresponds to an unlikely speed 
of 60 knots). This leads to the necessity of applying controls that obviously would be 
needed anyway for rough-water operation. A systematic study of the effectiveness of con- 
trols proportional to z, z, 0, and 6 could and should be made to arrive at the best combina- 
tion. In view of the shortness of time available for this study, computer runs were made only 
for a control function proportional to 6. An all movable control area of 300 ft? is considered 
to be actuated without time lag according to the equation 
Tas vAngtie=— 0. —wekGs 
When the gain factor k was varied to determine a suitable value, it was found that k = 1.0 is 
just enough at 30 knots to prevent divergent oscillations and that a value of k = 2.0 gives 
good control, requiring a tail angle of about twice the maximum pitch angle. Results of com- 
puter runs for this gain factor (& = 2) at different speeds are as follows, the values listed 
being the ratios of amplitudes of any cycle to that of the previous cycle: 
0.92 0.35 
0.30 critical damping 
critical damping critical damping 
It therefore appears possible to control the craft in calm water over the speed range inves- 
tigated by means that are quite reasonable. A more thorough analysis that involves the use 
of z, z, and 0 may well reveal a still more effective means to achieve controllability. 
Sensitivity of the Analysis, and Recommendations Regarding Stability and Control — 
Runs were made on an analog computer to investigate the sensitivity of the response of the 
646551 O—62——37 
