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Let us ignore the compressibility effects, i.e., assume that the flight Mach number
is below 0.5, then we can ignore the variation of Cma with Mach number so that
Cmu - 0. Then, for-such cases, the above condition reduces to
Recall that
Cnut - 0
Cma, = (Xcg - No)au
(6.100)
(6.101)
AIRPLANE RESPONSE AND CLOSED-LOOP CONTROL          555
Fourth Order System
-N  Shart-Period Approximatron
Fig. 6.7   Unit-step response of general aviation airplane: angle of attack and pitch
rate for first 10 s.
so that
xcg - No
(6.102)
   Thus, when the center of gravity coincides with the stick-fixed neutral point,
the rootjocus crosses the imaginary axis. For further aft locations of the center
of gravity, the root-locus moves into the right half of the s-plane, and the aircraft
becomes dynamically unstable as well. To have an idea of the nature of free
response of a statically unstable airplane, let us consider the general aviation
airplane once again with the assumption that the center of gravity is moved aft to
make it unstable in pitch. We know that the center of gravity has a direct effect
on the damping terms Cmq and Cm , through horizontal tail terms. To accoun
these effects, let C,,,u  - 0,07/rad and both Cm r and Cmq be reduced by 50%.
      The eigenvalues ofthis statically unstable version ofthe general aviation airp
were calculated as
      Ai - -3.1390
A2.3 = -0.2882 -1- j0.2212
A4 - 0.1464
for
ane
(6.103)
(6.104)
(6.105)
    Notice that the pair of complex roots A2.3 has a short-period-like damping and
phugoidlike frequency. This is the so-called third oscillatory mode.
    The free response to an initial disturbance in angle of attack of 5 deg is shown
in Fig. 6.9. We note that the disturbance in angle of attack decays initially because
I
556           PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
A,A : Short-Period Roots
                                                                                 
                                                                                 
                                                                                 
                                                               A ..- .  A A. J   
┏━┳━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃  ┃                    ┃\                                                   ┃
┃  ┃r        j(D        ┃                                                    ┃
　
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