Semisubmerged Ships 555 
Ry bed —(ad~ + b’e) 
> O FOR STABILITY 
< O FOR INSTABILITY 
acieun 
cae 
—————_..... 
FOR 40 knots 
(UNSTABLE) 
0 200 400 600 800 1000 
TAIL AREA IN FT 
Fig. 16. Variation of the condition for oscillatory stability, Ry 
upon whether the sign of the real part of this complex root is negative or positive. Routh’s 
discriminant (Rj) provides the following relationship, which the coefficients must satisfy 
for stability. 
Riosn beda= (ad? .b2eyachw 0 (16) 
It can be seen that the quantities d and e, which represent coupling of the free surface ef- 
fect with the body damping and static force rates must not become overly positive or the 
product bed will be overcome and oscillatory instability will result. Inspection of Fig. 16 
shows that Routh’s discriminant is positive at 60 knots for tail areas greater than 200 ft”, 
but that even a tail area of 1000 ft? is insufficient at 40 knots. The loss of stability with 
decreasing speed is due mostly to the large change in M, which, with some assistance from 
Z,, produces a 12-fold change in the coefficient e, thus valine it much more positive (even 
for a tail area of 300 ft”) and resulting in a negative value for Routh’s discriminant. It is 
also to be noted that increasing b, which is really the deep-water effective damping coef- 
ficient (depending on Z,, and uM), will cause oscillatory divergence. This is also a 
