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approximations as
Asp = -2.5060+ j2.5717
Ap, = -0.00611- j0.2616
(6.87)
(6.88)
    Comparing these eigenvalues with those of the complete fourth-order system
obtained earlier, we observe that the short-period approximation is quite satisfac-
tory for the prediction of free response of the general aviation aircraft  whereas the
frequency of phugoid motion is slightly in error. In general, for typical statically
stable aircraft, this type of comparison is obtained.
   The free responses of the general aviation airplane based on complete fourth-
order system and based on short-period and phugoid approximations with assumed
initial conditions Acy - 5 deg and u = 0.1 are shown in Fig. 6.4. The MATiLABl
was used for these calculations. We observe that the disturbances in angle of attack
and pitch rate q decay quickly and come close to zero within 3-5 s, whereas the
disturbances in forward velocity and pitch angle persist for a long time and decay
slowly. In other words, disturbances in angle of attack and pitch rate decay rapidly
during the short-period mode. During the phugoid mode, which continues after
the decay of the short-period mode, the value of angle of attack nearly remains
constant, and the pitch rate is close to zero.
     Next, let us examine the accuracy of the short-period and the phugoid approx-
imations for forced response. For this purpose, we have computed the unit-step
AIRPLANE RESPONSE AND CLOSED-LOOP CONTROL          551
Fourth Order System
Phugoid Approximation
10
g
!
-10
                            A/ \j~ f.,.\
0    20    40    60    80
100   120   140   160   180   200
time
F:ig. 6.4    Longitudinal response of general aviation :urplane.
response ofthe general aviation zurplane, and the results are presented in Figs. 6.5-
6.7. It is interesting to observe that the step responses differ considerably. The
short-period response agrees well with that of the complete fourth-order system
initially, but the steady-state values differ. The short-period approximation results
in a quick decay of the oscillator}r motion, whereas the complete system continues
to oscillate for amuchlonger time. The step response based on phugoid approxirna-
tion differs considerably from that for the complete system. There is a significant
difference in the transient as well as steady-state values. rrhus, for the general
aviation airplane, the forced response based on short-period approximation is sat-
isfactory, whereas the phugoid approximation is not adequate at all. Therefore, it
is always a good practice to check any controllaw design based on such approxi-
mations by a simulation of the complete system using the same control law. If the
two simulations are in fair agreement, then the design is satisfactory. If not, the
exercise has to be repeated using the complete system instead of a reduced-order
system based on either short-period or phugoid approximations.
6.2.4   Effect of Static StabLfity on Longitudinal Response
   When the center of gravity moves aft, the static stability level of the airplane
decreases. When it goes aft of the stick-fixed neutral point, the aircraft becomes
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552           PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Frea Response of Genaral Aviation Airplane
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