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- sin ct sin 4 cos p + s:in p cos 4
sin ar cos 4 cos p - sin p sin 4
330 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
pcff = sin-l
( UV.)
= sin-l(sin ct sin 4)
(4.47)
If ~ : 90 deg, then aefr -. 0 and Peff -. af. In other words, when the bank angle is
90 deg, all the angle of attack.gets converted to sideslip.
4.2.6 Euler Angle Rates
One ofthe problemsin flight dynamicsis to compute the time history of the Euler
angles.However, such computations need aknowledge ofthe Euler angle rates 4, 0,
and j,, which are not directly measured or are not available. What are generally
available are the angular velocity components p, q,r, which are the body axes
components of the angular velocity of the vehicle with respect to an inertial axes
system.ln other words,we are given P, q, r in a body-fixed system and are asked to
find the Euler angle rates lt-, 0, and ~. The angular velocity components p, q, r in
the body-fixed system may be available either from onboard measurements using
rate gyros or may be derived from a solution of the equations of motion*
Let us refer to Fig. 4.3 with an understanding that Oxi corresponds to Oxi,
OYi to Oyt, and Ozi to Ozi.With this understanding, we observe that the angular
velocity vector ~ is directed along the Ozi or Oz: axis, the angular velocit3r vector
0 along the Oyi or yi" axis, and the angular velocit)r vector 4 is directed along
the Oxz or Ox:." axis. Based on this information, we can determine the relations
between body axes rates p, q, r and the Euler angle rates Vr, 0, and @ as follows.
To begin with, consider the p vector.lt has to be transformed from the Ox:.y:z:
system to the Oxbyir,zb system, and the corresponding transformation matrix is the
matrix product AB so that
~b = AB [,;.] (4.48)
where l/ b denotes the vector ~ resolved in the.Oxbyt,zb system.
Next, consider the angular velocitjr vector 0, which is directed along the Oy:
axis of the Oxrt'yt'z:! system (Fig. 4.3).
(4.49)
Finally, we consider the angular velocity vector 4, which is directed along the
Oxb axis of the Oxhybzb system (Fig. 4.3) so that
4b =
(4.50)
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 331
The angular velocity vectorin body axes system Oxbybzb is given by
Therefore,
tOl.,b =
[-q-j,]
(4.51)
[p:] =AB [,;.] +A [:;] + ~~] (4.52)
Substituting for matrices A and B from Eqs. (4.14) and (4.20), we get
O
[pc;f = [j:S4:SSS,::% jos4
-sin 4
1 O 01
+Ocos~ ,l[;jl+[%j] (4.53)
0 -sin~ ims%j '
Simplif)/ing, we obtain
so that
[q-j,] = [: _SS:.4,
p = 4 - ~'sin0
q = O cos 4 + ~' sin ~ cos 0
r : jr cos ~ cos0 - 0 sin ~
~~i]
(4.54)
(4.55)
(4.56)
(4.57)
From Eqs. (4.55-4.57), we observe that very-often-used relations such as p = 4,
q = O, and r -. y. are true only when both the pitch angle 0 and bank angle 4 are
close to zero.
Let
1 0
Lco = 0 cos4
0 -sm~
(4.58)
-sin 0
sin CO:s%] U~]
cOs ~ cOs (
1/lOO:s%]
1i'~s2]
[%i]
332 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
so that
~~i] = ,.-' [-qj,] (4.59)
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