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  A  e_ =  0~695 S3 +0 8494 S2 +2~851s                     (6.163)
AO(s). _            -2.0112s - 3.9018
  Ai;t- = 0~695 S3+0~94s2+2~85~s                (6.164)
q(s)                   -2.0112 S2 ~ 3.9018 s
   - == -                                         (6.165)
A8e 0~695S3+08494S2+2~851s
The transfer functions of the general aviation airplane based on phugoid approxi-
mation [substitute in Eqs. (6.148) and (6.156)] are given by
u(s)      0.1455         (6.166)
A7ie= 4~3~4S2 02145s 03362
A6(s)   -0.7831s-0.0355      (6.167)
Z 8e =  4~3~4s-02~45s-0.3362
   We will be malang use of these transfer functions for Lhe design of stability
augmentation systems and autopilots later in this chapter.
6.2.6 LongitudinaIFrequencyResponse
co
o
a
AIRPLANE RESPONSE AND CLOSED-LOOP CONTROL         563
┏━━━━━┳━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃          ┃                ┃.".,'...,,.,..{...ll,                                     ┃
┣━━━━━╋━━━━━━━━┫                                                          ┃
┃-........ ┃. . . .. .... . ┃'i:.^<-..-::.. -"'\:.-::.'....  :. .. . .-.-'-' ...'..... ┃
┃          ┃                ┃               , ""'-.-.:. . '-'-'                        ┃
┣━━━━━┻━━━━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┫
┃                            ┃                                                          ┃
┗━━━━━━━━━━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛
          Fr~aquenq, rad/s
Frequency response: forward velocity
Frequency, rad/s
Fig. 6.10    Longitudinal frequency response of the general aviation airplane: forward
velocit}r.
expected that a human pilot can control flight path variables that respond to input
frequencies within this range. The autopilots t:/ave a higher bandwidth. A typical
value of autopilot bandwidth is about 20-25 racUs.
     To illustrate the nature oflongitudinal frequency response of conventional, stat-
ically stable airplanes, we have presented the longitudinal frequency response of
the general aviation airplane based on complete fourth-order transfer function in
Figs. 6.10-6.12. Here, Gu, Gcy, and Go are Lhe longitudinal transfer functions of
the general aviation airplane given in Eqs. (6.160-6.162). From the magnitude
plot of Fig. 6.10, we observe that IGul attains a peak value around the phugoid
frequency COph and then drops off rapidly. The magnitude crossover frequency (the
frequency where the magnitude is O db) for IG ulis about 0.5 rad/s. This implies that
elevator input with frequencies beyond 0.5 rad/s have little or no effect on the for-
ward speed. For example at co -. 3.6 rad/s, which corresponds to the short- period
　
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