556 Edward, V. Lewis and John P: Breslin 
surprising result, which stems from the coupling of the body coefficients with the free sur- 
face spring terms (M, and Z,). Thus, making the body overly stable for deep operation pro- 
vides a very stiff effective spring coefficient (e), which appears to be most responsible for 
divergent oscillatory oscillations. 
Results of Analog Studies 
Exploration of the influence of fin area, distribution of fin area, the sensitivity to cer- 
tain coefficients, and the effectiveness of control can quickly be assessed by an analog com- 
puter. The results of such studies are discussed in the following subsections. 
Responses with Fixed Fin Areas — Results of the analog studies of responses with fixed 
fin areas can be friefly summarized as in Table 3. 
Table 3 
Analog Computer Results for Fixed Fin Areas 
Speed All Fin 3/4-Area Aft, 1/2-Area Aft, 
(knots) Area Aft 1/4-Area Forward 1/2-Area Forward 
Fin Area = 300 ft? 
30 Divergent 
oscillations 
40 Divergent 
oscillations 
60 Damped 
oscillations 
Fin Area = 1000 ft? 
40 Slightly divergent Damped Divergent 
oscillations oscillations oscillations 
60 Lightly damped Heavily damped 
oscillations oscillations 
Since the case of 300 ft? of fin area has been discussed, the following remarks will be 
directed toward giving some rationalization of the curious result of very slight stability at 
60 knots with 1000 ft? of fin area aft as compared with very marked stability when this area 
is split into half forward and half aft. To begin with, this large area was selected as an 
intuitive attempt to stabilize against the oscillatory instability at 30 knots without regard to 
its implications on the stability of the deeply operating submarine. An inspection of the 
criterion of stability for deep operation, 
M,Z, - M,(m+Z,) > 0, 
which is essentially the quantity c, shows that by adding too much tail fin, m + Z, can be 
made negative (by making Z, so negative that it overwhelms m) and, at the same time, 
