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Plot the Euler angle time history for an airplane performing a velocity vector
roll with 1) LY - 30 deg, 2) a - 45 deg, and 3) ce -. 60 deg. Assume that the
angular velocity S2 about the velocity vectorin each case is 25 deg/s.
Solution The first step is to select a reference frame that remains fixed in
space and with respect to which the Euler angles are measured. Let xoYozo be such
a reference frame as shown -in Fig. 4.8. The Oxo axis is assumed to coincide with
the velocity vector Uo, the OYo axis points to the rightin the horizontal plane, and
Ozo points vertically downward to form a right-hand system.
At t - 0, the aircraftis assumed to be oriented as shown in Fig. 4.8 so that the
initial values of the Euler angles of the body axes system Oxbybzb are rjr (0) - 0,
0(0) = ct, and ~(0) = O.
zo
Fig. 4 8 Airplane in a velocit)r vector roll.
""
.:i
Ill:-:"
cos S2eti
-. -siri CZetl
0
[X:E::]
350 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
We have
p -. s-z cos a
q -.0
r - S2 sina
Because the angle of attack remains constant during the velocity vector roll, the
values of p, q, r will also remain constant.
With q = 0, the Euler angle rates as given by Eqs. (4.66-4.68) are
4 = p+r tanOcos4
0 - -r sin4
~ - r sec0 cos4
An integration of these equations for the above initial values was performed
using the MATLABrM (Ref. 5) routine ODE45, and the results are shown in
Fig. 4.9. We observe that the roll angle continuously increases with time.ln view
of the restrictions as shown in Eqs. (4.1-4.3), the roll angle 4 has a sawtooth
variation.
07}
0. 0
-200
Euler Angles in Velocity Vector Roll, Omega= 25 deg/s
I ---t~ J
(: ::. -/::- ;1 ;-
┏━━━━━━━━━━━━━━┓
┃ : . ,P---. ┃
┗━━━━━━━━━━━━━━┛
o . 5 10 15 20 25 30
-fhg. 4.9 Euler angle time histories in velocit}r \rector rolls.
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 351
a
Fig. 4.10 Spinning airplane of Examp,le 4.3.
Example 4.3
zo
For a spinrung airplane as shown in Fig. 4.10, the angular velocity components
in the body axes are given by
p = s? cOs tx
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
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