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(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)
The transformation matrix Lo is not an orthogonal matrix because the vectors
lv, O, and 4 are not mutually perpendicular. Hence, the inverse is not equal to the
transpose. Therefore, we have to compute L~l in the usual way as follows:
~:l = (A(1,))adj( ~) (4.60)
The determinant A(L) of matrix Lcn is given by
A(L) = COS2 ~ cos O + S1112 ~ COS 0 (4.61)
N cos O (4.62)
N
The adjoint of matrix L~ is given by
'cos0 sin0 sin~ cos@ sin
adj(Lo,)= O cos~cos0 -s,.,'S,] (4.63)
0 sin~ cos4
so that
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