The rate change of submergence d is given by 
z-Tf + (z -1r) d(cos 8 - usin 0) 
eo" ee 
cos 6 - usin 0 (cos @ - v sin 6)2 ot 
Since immersion (z-r) is always small in the valid range of the previously described 
expression, the relationship can be further simplified to 
WG gee 
~ cos 6 -vsin 8 
and 
(z -1) 
m, ~ k,tpb(n/2 cot 8) ~—— 
The expansion of the integral expression for the hydrodynamic moment in pitch 
follows the procedure used for the vertical force. The results are summarized as follows 
Fo = -1,0 + Q, cos O Zog - Q,4Z¢g sin 8 - kg cos 8) 
dw, z 
~ fm, cose =? kat + [m6 sin 0 w, dé 
Q ! Q 
+ [ vingeat + [ocybveede 
Q Q 
+m,UVé| + [m,vuee 
stern Q 
uf dw, 
st; m,V —— sin @éd 
us dw, 
- / m,U = cos0@éd 
J m,U Gp cos 0 Edt 
+ [aoe cos 6 €dé 
Q 
The only additional moments are the buoyancy moments. All other moments are considered 
to be zero for the specific problem considered in this report. 
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