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of reference OxoYozo and a moving axes system Oxiyizi as shown in Fig. 4.15.
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 361
x.,xo
t=o
yo
ri
zo
t. At
Fig. 4.15 Scheinaticillustration ofthe moving axes theorem.
yl
yo
Let eo be the angular velocity of the OxiYizi system with respect to the OxoYoZo
system but having components in the OxiYiZi system. Assume that at t - 0 the
OxiYizi system coincides with the OxoYozo system. Let a particle P move with
a constant velocitjr uo along the Oxo axis. At t = At, the parrlhcle P will still have
the same velocity uo with respect to OxoYozo so that the acceleration measured by
an observer stationed at the origin of the OxoYozo system is zero.
Now let us find out what an observer stationed at the origin of the moving
coordinate system OxiyiZi has measured. At t - 0, he will also record ui - uo
and vi - wi - 0. At t - At, he will have ui - uo cos cot, vi - - uo sin c.ot, and
WI = 0. Thus, according to him, the particle P has the accelerations
so that
Ul = ^hm uo(coscoAt - 1) 0
At-+0 At
. uo sin roAt - O
-- : -uotjo
Vl = ~li/io-- A~
Wl -0
ddV, ), = - jluOco
Thus, the acceleration measured in the moving coordinate
from that recorded in the inertial reference system. According
theorem,
(ddV)o = (dd )i+coo,i X (V)l
We have
(V)1 - ZIU1 + /lvl + kiwi
- ki co
(4.227)
(4.228)
(4.229)
(4.230)
system is different
to the moving axes
(4.231)
(4.232)
(4.233)
362 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
We have ui - uo and v\ -- wi - O so that
7oAi x cv)i - Jiuoco
Substituting in Eq. (4.231), we get
ddV, )O = --Jtluoco + J*luoco
Thus, the theorem holds.
-O
4-3.2 Expressions for Velocityan.d Acceleration
(4.234)
(4.235)
Let us consider the motion of a rigid body as observed in various coordinate
systems as shown in Fig. 4.16. Let xiy,zi be a nonrotating reference system fixed
at the center of the rotating Earth. Let xbybzb be a coordinate system fixed to the
body and moving with it. Let xeyeze be a system fixed to the surface of the Earth
and located directly below the body at t = O (navigational system). The OxeYeze
system rotates with the Earth and has a constant angular velocity S2e with respect
to the OxiYtz, system. Here, we assume that xiy,zt serves as an inertial frame of
components in the Oxt ybZb system.
cra
lb
Fig. 4.16 Inertial, narr,gational, and body axes systems.
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATfVES 363
Then,
Rr - Re+RO+rb , ' (4.236)
(ddR ). = dd (Re + Ro + r+~,)
= (dR )t+(ddR ).+(dd ).
Using the moving axis theorem shown in Eq. (4.226), we have
(ddR ).= (dc~R )e + S2e x Re
. = (2e x Re
(4.237)
(4.23 8)
(4.239)
(4.240)
where ?2e iS the angular 'velocity of the Earth-fixed OxeYeze system with respect
to the Oxiy,zi system but having components in the OxeYeze system as given by
S?,e = le S-Ze COS A, - keSZe siri )L
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