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= t,z (Vo)e
(4.285)
= , (:V, )e . (4.286)
Using the moving axes theorem, we have
(ddV, )e = (ddJ, )b+Wl.b X Vo (4.287)
With (Vo)b = lb U + ib V + kb W and 7.o~,b = t70lb = 7b p + 7bq + kbr, we have
(ddV, ), = 7b U + jb V + kb W (4.288)
COe,b X Vo = 7b(q W - Vr) - jb(p W ~ Ur) + kb(p V - Uq) (4.289)
With
F = Zb Fx + lb Fy + kb Fz (4.290)
we have the following force equations for aircraft motion in Cartesian form:
Fx =m(U+qW -rV) (4.291)
Fy -m(V +rU - pW) (4.292)
Fz -m(W + pV -qU) (4.293)
368 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Theorem on angular momentum. Consider the motion of a particle P of
mass 8m with respect to the Earth-centered inertial frame ofreference XrYtZt (see
Fig. 4.16). Let XbybZb denote the body-fixed axes system and let the orig:in of the
body-axes system be located at the center of gravity of the body. From Newton's
first law of motion,
8F1 - 8m Vr (4.294)
- 8m Ri
.= 8m(Ro + rb)
. '. -
Summing over the entire body,
Ft = E8F:
= E8m Ro + E 8mrb
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