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- (-keRe lr2~ (4.267)
= ReS2~ COS } (le sin A + ke cos A) (4.268)
366 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
The vector cross product
s2c X Ve = ref2e Voy sin A - jc(S2c Voz cos A + Q,e Vox sin A)
+ ke S2e Voy COS A. (4.269)
Substituting and rearranging, we obtain the following equations ofmotionin Carte-
sian form:
Re S2~ cos }.sin A + Xe + 2S2e Voy sin A = Fx.e (4.270)
ye - 2(2e( Voz cos A + Vox sin A) = Fy, (4.271)
Re C2~ COS2 A. + Ze + 2<2e Voy COS A, = Fz.e (4.272)
where
F = le F.r,e + Je Fy,e + ke Fz.e (4.273)
The above system ofequations describes the motion ofa point mass with the only
assumption that the orbital motion of the Earth around the sun is ignored. For the
majority of the fiight dynamics problems, this assumption is usually satisfactory.
However, for interplanetary motions like a mission to the planet Mars or other
distant planets, the orbital motion of the Earth has to be considered. The interested
reader may refer to a standard text on astrodynamics for this purpose.6
4.3.3 Force Equations for Aircraft Motion
In the study of aircraft flight dynamics,it is usual to ignore the rotation of the
Earth about its own axis.ln other words, an Earth-fixed axes system such as the
navigational system XeYeZe is assumed to bc an inertial frame ofreference.ln doing
so, weignore the terms S2<x :Qe X Re and 2(2e x Voe.ln other words, we assume
eob. b = C:Oeb,b and ( Vo)t - ( Vo)e. The error introduced by these assumptions can be
estimated as follows.
Consider an aircraft moving at a flight velocity of 1000 mph-(1466.37 ft/s) at sea
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