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n
zi , L
o
a)
k--.
┏━━━━━━━┓
┃ \ ┃
┃ \ ┃
┃ \ ┃
┗━━━━━━━┛
st
PLani,
Plane o. Rotation
Fig.4.7 Angular velocityin a navigationalsystem.
-iS,:os: ]
336 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
We have
wi,b = TteCO;,b
Using Eq. (4.30) for 7,e, we obtain
(4.88)
-sin A cos L -sin A,sin L cos A
Wf.b= -siriL cosL O,AAl[_:s;os:l ,489,
-cos A,cos L -cos A sin L -sin A
= [-,j-:}'o{~. : ] (4.90)
Here, coZb is.the angular velocity of the spacecraft with respect to theinertial system
and having components in the navigational system.
[p7]=AB[_B]+A[2jl+[:;] (4.91)
Substituting for matrices A and B from Eqs. (4.14) and (4.20) and simplifying, we
obtain
[q:] = [:
From Eq. (4.92), we get
0 ' +sin a
cos 4 -sin 4 cos ct
-sin ~ -cos~cos ct
0
cos 4
-sin 4
p =4+Bsina
q = -p sin~cos a +cycos4
r = -p cos ~ cos ce - a sin ~
Ci
(4.92)
(4.93)
(4.94)
(4.95)
(4.96)
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 337
We note that (T3)-1 j/ (Tu~)' because the angular velocity 'vectors a, 8, and 4 are
not mutually perpendicular. Therefore, the matrix T~ is not an orthogonal matrix,
and the inverse is not equal to the transpose. Therefore, (T~)-l was computed in
the usual fashion using Eq, (4.60) and is given by
Then,
Tbw = (T~)-l = [:
sin 4 tan ct
cos @
sin 4 sec ct
4 = p+tana(qsin~+r cos~)
a = q cos ~ -r sin ~
p = -seca(q sin~ +r cos~)
(4.97)
(4.98)
(4.99)
(4.100)
4.2.7 Method of Direction Cosine Matrix
Consider a vector A - iAx + jAy + 7cAz in the Oxyz system. Let 81, 82, and
83 be the angles the vector A makes with the x, y, and z axes, respectrvely. Then,
we have
so that
Ax
cos 81 = IAj (4.101)
(4.102)
(4.103)
COS2 8i +COS2 82 + COS2 83 = 1 (4.104)
The numbers cos 81, cos 82, and cos 83 are called the direction cosines of the vector
A with respect to the Oxyz axes system.
Now consider the transformation of a vector from one coordinate system to
another using the direction cosines. Suppose we want to transform a vector given
in the OxiYizi systeminto Ox2Y222.Let
l2 = Ciiii + C12/1 + Cl3kl
l/2 = C2111 + C22 71 + C23ki
k2 = C3111 + C32]1 + C33kl
(4.105)
(4.1'06)
(4.107)
'0-4vl
cos 4 sec cej
,os 82 = IAAj
,os83 = fAAI
-:,js',:;,::::]
424 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
a)
Mach Number
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