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 Eder AWbs and Ratosln Spln.Omega-30 alpha1 30
a) Alpha = 30 deg
 Euler Angles and Rates In Spn, Omega=30 alpha;  60
                             b) Alpha = 60 deg
Ng. 4.11    Euler angles and Euler angle rates in spin.
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 353
the body axis Oxb (Fig. 4.lla). On the other hand, at high angles of attack, it will
be mostly yawing motion as observed for cr -. 60 deg (Fig. 4.llb).
                                                 Example 4.4
     An aircraft model is tested in a Iow-speed wind tunnel at an angle of attack
of 20 deg, sideslip of 10 deg; and a bank angle of 10 deg. An internal strain
gage balance was used to measure the aerodynamic forces acting on the model,
 which gives components of force in the body axes system. The measurements are
 Fx = 21.7 lb, Fy = -33.0 lb, and Fz = -91 lb. Determine 1) the transformation
matrix Tbw and 2) the lift  drag, and side forces acting on the model.
                          -cos ct sin )3
-sin  p sin a sin 4 + cos p cos 4
-sin tS sin cr cos 4 - cos p sin cr
With cv = 20 deg,,B = 10 deg, and @ -. 30 deg, we get
Then,
      0,9254  0.3188
Tbw =   -0.1632  0.8232
      -0.3420 0.4698
[F-',;w:] = Tbw [--'~O]
so that F.rw - -9.0809 lb, Fyw = 4.6226 lb, and Fzw = -96.9833 lb or lift
/ ~ 96.9833 lb, drag D - 9.0809 lb, and side force Y  - 4.6226 lb.
                                          Example 4.5
   For the spinning airplane of Example 4.3, determine the Euler angle history
using 1) Euler angles, 2) the method of direction cosines, and 3) quatenuons for
a = 30 deg, 45 deg, and 60 deg.
      So/ution.     The reference axes system OxoYozo with respect to which the Euler
angles are measured is assumed to coincide with the body axes system at t -- O.
"
't:l
     So/ution.    The forces Fr, Fy, and Fz measured by the balance are with respect
to the body-fixed axes system, whereas the lift, drag, and pitching moment are with
respect to the wind axes system. Therefore, we need the transformation matrix Tbw.
    From Eq. (4.37), we have    .
-8
~
t;
c:l
6
r:
 o
6
€
j::,::~]
                                 cos a cos p          .
T3 =   sinasin4.cosp+sin-8cos4
                sin or cos4 cos f/ - sin  p sin ~
6
oI
u
a
u
a
0.2049 -1
-0.3882
0.8138 _1
3]
f
-=
c:
a
354                PERFORMANCE, STABILI-fY, DYNAMICS, AND CONTROL
Subsequently, the reference axes system xoYoZo remains fixed in space. This means
that we will have a different reference axes system for each value of a. With these
assumpOons,  k(o) = o(o) = 4(o) = 0 for each case. We have
p - s2 cos a
q-0
 r - 02 sinaf
Because cr is a held constant for each case, p = q = r = 0.
   1) Euler angles: the Euler angle rates are given by Eqs. (4.66-4.68). For this
case, these equations reduce to
4 = p+r tanOcos4
     0 = -r sin4
    j/ : r sec 0 cos4
The MATLABs code ODE45 was used for numericalintegration ofthe above Euler
angle rate equations. Whenever the value of any Euler angle exceeded 180 deg, it
was reset equal to -180 deg.
    For ct = 45 deg, the pitch angle 0 approaches -90 deg, and we encounter the
singularity present in Euler angle rate equations. To work around this difficulty,
　
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