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                   ,    W = Uo(sinacos~cosp - sinpsin~)
The angle of attack and sideslip are customaril:         ined as
          a=_-,( )define
(4.39)
(4.40)
(4.41)
(4.42)
(4.43)
(4.44)
                                        p = si.-, (j/)                               (4.45)
With U, V, W given by the Eqs. (4.41-4.43), we find that the above definitions
of a and p hold only when the bank angle # -. 0.If ~ + 0, part of the angle of
attack gets converted to sideslip. As a result, the effective angle of attack aet:r will
be smaller than the given angle of attack a, and the effective sideslip peff will riot
be zero. This can be visualized as follows.
     Consider an aircraft model mounted in a wind-tunnel test section. Let ct be the
angle between the modellongitudinal axis (zero-lift line) and the tunnel axis and
4 denote the bank angle about the modellongitudinal axis. Assume that the model
is not given sideslip or p - 0. .
    With p = O, using Eqs. (4.41-4.45), we have
Cteff = taDc-l
(:)
= tari-l(tari ce cos 4)                                     (4.46)

jl:}il~a~]
                             cos a cos p
-     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).
　
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