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q=0
r -. r2 sin ct
where S-2 is the angular velocity about the spin axis, which is assumed to be vertical
and passing through the center of gravity. Plot the Euler angle time history for the
first 25 s and two values of ct -. 30 deg and 60 deg. Assume Q -. 30 deg/s.
So/ution. Because q = O, the airplane has zero pitch rate about the body axes.
It experiences only rolling and yawing motion about the body axes.
We select the reference frame xoYoZo with respect to which the Euler angles are
calculated to coincide with the body axes system at t - 0. For t > 0, the reference
frame xoYozo remains fixed in space, whereas the body-fixed axes system xbybzb
system rotates with the airplane.ln view ofthis, we have /r(0) _ 0(0) : ~(0) = 0.
Note that for each case of ct - 30 deg and 60 deg, the initial orientation of the
xoYozo axes system is different.
With these initial conditions and using Eqs. (4.66-4,68), the Euler angle time
history was obtained as shown in Figs. 4.lla and 4.llb. At low angles of attack,
as in the case of ct - 30 deg, the spinning motion is predominantly the roll about
o)
o)
D
s
~
.r:
352 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
- EulcrAngles,dql
---_ EulerAnglcs Rates,deg
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.
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