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where S2e = l~el, the magnitude of l2e. Furthermore,
(ddR )t = (ddR )e + (2e X Ro
= Vo + g3e x Ro
(4.241)
(4.242)
(4.243)
Here, V o = (dRo/dt)e is the velocity ofthe body with respect to the inertial systcm
having components in the Oxeyeze system.
(dd ). = (dd )b+(-t3rb X 7b (4.244)
= t3{.,b x rb (4.245)
The first term on the right-hand side of Eq. (4.244) vanishes because we have
assumed that the body is rigid. Then,
(ddR ): = ~e X --e + -,~+ g2e X Ro+O}:.b x rb (4.246)
(ddR ). = dd (S2e X Re)le + g2e X f2. x f/e + (ddV, )e + ?2e X veA
~l
+ ?2e x V~ + g2e X 62e x Ro + [dd (o3l.b X 7,)]b + (3J.f, X (CObt, X rt)
(4.247)
364 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
We note that the first term on the right-hand side is zero because both S2e and Re
are constant vectors in the OxeYeze system. Then,
(dcfR ). = ~e X (~e X Re) + (ao)e+2Qe x VZ
+ (2e X (l2c x Ro) + (dd )b X 7b + c~Obb'X tob.,l, X 7b (4.248)
= g2~ X Qe x (Re + Ro) + (ao)e+2t2e x V~+ (d )h .X 7b
+ cOlb X 03lb x rb (4.249)
where (ao)e = (d V o/dt)e. Furthermore, we have assumed that the body is rigid so
that (drb/dt)b - 0.
In Eq. (4.249), various terms have the following interpretation:
S2e x S2e x (Re + Ro): centripetal acceleration at the origin of the body-fixed
system due to rotation of the Earth.
(ao)e: acceleration of the body relative to an Earth-fixed system OxeYeze.
292e x V:,: Coriolis acceleration observedin an Earth-fixed system.
.(dC~Obb/dt)b x rb: apparent acceleration observed in the body-fixed system due
to a tune rate of change of the angular velocity of the body.
CO:bb X co~ X rb: apparent acceleration ofthe particle due to the angular velocity
of the mowng axes system and recorded in the body fixed system.
Force equations. Multiplying both sides of Eq. (4.249) by 8m, mass of the
particle, we obtain
8m d2R_ ~ 8m<2e x S2e X (Re + Ro) +8m(ao)e + 28rriSle x'V:
8m CTt2 :
+8m(dd,),xl,+8mo3lbx(c3{.bxr_t,) . (4.250)
~ 8Fz (4.251)
where 8Ft is the net external force acting on the particle P of mass 8m.
Summing over the entire body,
>:j: 8F, = E8m?2e x C2e X (Re + Ro) + ~- 8m(ao)c +2E 8m:Qe x vo
Let
+ X) 8m(dc~ x b) + ~_ 8m(colb x colb X rb) (4.252)
:8m =m
~- 8Ft -. F
(4.253)
(4.254)
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 365
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