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2468
10    12    14    16    18    20
Ume
Fig. 6.4    Longitudinal response of general a\riation airplane, continued.
statically unstable. In the s-plane, the short-period and phugoid roots move to-
wards the real axis as shown in Fig. 6.8, Here, AA and B B denote the short- period
and phugoid roots for a stable location of the center of gravity. When the center
of gravity moves forward, the roots move onto the real axis, and the conventional
short-period and phugoid approximations break down. The right moving branch
of the short-period mode meets the left moving branch of the phugoid mode and a
new oscillator)r mode emerges. This mode has the short-period like damping and
phugoid like frequency. This mode is sometimes known as the third oscillatory
mode. At the same time, the right moving branch of the phugoid mode crosses the
 imaginary axis and moves into the right half of the s-plane. This indicates that the
airplane will exhibit an exponential instability in pitch.
     The condition when the real root crosses the imaginary axis and moves into the
right half of the s-plane can be determined as follows.
   According to Routh's criterion, the necessary condition for stability is that all
coefficients ofthe characteristic polynomialmust be positive or have the same sign.
When any one of the coefficients becomes negative while the rest are positive, then
 the necessary eonditionis violated and the system becomes unstable. By examining
the expression of all the coefficients of the characteristic polynomial, we observe
that the coefficient that is most likely to change sign duve"tlo a change in static
stability level is the coefficient E8 as given by Eq. (6.25). Therefore, the condition
AIRPLANE RESPONSE AND CLOSED-LOOP CONTROL         553
Phugoid Approximation
Fig. 6.5    Unit-step response of general aviation airplane: forward \relocity and pitch
angle.
for the root-locus to cross the imaginary axis is
E8 = -Cxa Cm i Czq Cl  - CxO(Czu Cmcr - Cza Cmu)
     - 0 .                                                                          (6.89)
Cmr - Cmu(Cx0Czcr - Cxa CzqCl)
                             Cxo Czu
We can express the derivative Cm, as
Cmri =
acm
a( ^y\ )
(6.90)
(6.91)
(6.92)
(6.93)
U,
o
o
m
cn
c
<
s:
o
:-
a-
c
m
cn
c
(o
L:
o
- M acm
  aM
= Mty aac,n rM
554           PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Short-Period Approxinuuion
Fig. 6.6   Unit-step response ot'general aviation airplane: angle of attack and pitch
rate.
Furthermore, we have
Substituting, we get
 Cxcr - CL - CDu
 Cx0 = -CL COS 0o
    Czq = -CLq
Czri - -2CL - CLu
Cza - -Cu N CD
(6.94)
(6.95)
(6.96)
(6.97)
(6.98)
Cmar =  Cmu[C~ COS Oo(CLct + CD) + (CL - CDa,)CLq CI1                 (6.99)
                         ~L~S 0o~2CL+~L t)
　
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