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┃        ':       ': o      ':                                                                                                                                       ┃
┃    ...i.ii                                                                                                                                                         ┃
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E
1.5     2     2.5     3     3.5     4
      Mach Number
Fig.3.61   Variation ofx"e and C,na with Mach number.
243
  From Figs. 3.60 and 3.61 we observe that present calculations are in quite
good agreement with APAS results except around M - 1, where both methods are
approximate.
                                     Example 3.3
    An aircraft has the following data: xcg  - 0.3, Xac - 0.24, CL.w  = O.lO((Xn +
2.5), CL.max = 1.2, C   c.w = 0.06, e = 0.3a, Cnf = 0.05 + O.lCL, ar = 0.08/deg,
th = 0.9, Vi  = 0.6, rw = O,/r = 2deg, Cha  = -0.002]deg,Ch&  = -0.003/deg,
and r - 0.20.
     Determine l) the angle of attack in steady level flight if the elevator is locked in
neutral position, 2) the permissible most forward position of the center of gravity
if the maximum up elevator defiection is limited to 30 deg, and 3) stick-free neutral
point and stick-free margin.
where
Cm = Cmo + (:C~ ),,x C~
Cmo = Cm*ac + Cmjo - ai V l r7t(cL*n.oL - iw + /t)
- 0.06 + 0.05 - 0.08 * 0.6 * 0.9 * (-2.5 - 0+ 2)
- 0.1316
338                PERFORMANCE, STABfLITY, DYNAMICS, AND CONTROL
          C12 C
[{IlI2;] =  'g"'  -, 22  'g:] [::":]
          C32 C
(4.108)
where Cii, C12, C13 are the direction cosines of the urut vector 12 with respect to
the OxiYizi system and so on. We observe that
C21
C31
   :  :
- I2 . II
    :   >
- J2 '11
=k2 11
C12 - l2 . Ji
C22 : j2 . jl
C32 = k2 . 71
C13 - 22 . ki
C23 -  j2 . ki
C33 = 7C2 . ki
(4.109)
(4.110)
(4.111)
where the " - " denotes thescalar product  Thus,knowing allthe elements Czj, /,  j  =
1, 3 of the direction cosine matrix, we can transform a vector from the OxiYiZi to
the Ox2Y2z2 system as follows:
Let
so that
or
          C12 C
[X:2;l= Cg"' 22 ":l[X;
         C32 C
cf =
Xz =
[X;2;]
X, = [x:"]
C12
C22
C32
X2 = C21Xl
(4.112)
(4.113)
(4.114)
(4.115)
(4.116)
(4.117)
(4.118)
　
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