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accelerations measured in the moving coordinate system are not the accelerations
with respect to an inertial frame of reference. Fortunately, this problem is a simpler
one to solve, and the following theorem of vector analysis helps us obtain the
accelerations with respect to an inertial frame of reference given the accelerations
in a moving coordinate system.
4.3.7 Moving Axes Theorem
Let Ab be a vector observed in the moving axes system Oxt,ybzb and (dAldt)b
be the time rate of change of Ab recorded in the moving axes system. Furthermore,
let tOlb be the angular velocity of the moving axes system measured with respect
to an mertial system but having components in the body axes system. As said
before, the order of the subscripts and superscript is as follows: the subscripts
/, b have the meaning: of the body axes with respect to the inertial axes system
and the superscript b means that the vector /3 iS resolved or has components in
the body_axes system. The problem is to deternune the time rate of change of a
d_
vector A in the inertial frame of reference OxiyLz, given its time rate of change
in the body axes system Oxbybzt . This can be accomplished using the following
theorem known as the moving axes theorem:
(-,,)t = ( d ) +COl,b X Ab
(4.226)
To understand the concept of moving axes theorem, consider an inertial frame
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
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