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(6.52)
(6.53)
Comparing this equation with the standard form of a characteristic equation of a
second-order system shown in Eq. (6.34), we get
LOn ~
< = 2B
-(c:,  + j,., (Cmq + Cma))
2   q Crm/
       I    lrrl
(6.54)
(6.55)
(6.56)
(6.57)
   From the above analysis we observe that 1) the frequency of the short-period
mode depends directly on the magnitude of the static stabjility parameter Cma
and 2) damping of th~ short-period mode directly depends on the damping-in-
pitch derivative Cmq and the acceleration derivative Cma. We know thaL the static
stability parameter Cmcr iS directly related to the center of gravity position. As the
center of gravity moves forward, Cmcr decreases (becomes more stable), and the
frequency of the short-period mode also increases. Conversely, if the center of
gravity moves aft, Cma increases (becomes less stable), and the frequency of the
short-period mode decreases. At one point, as we see later, when Cma  > 0, the
short-period mode breaks up into two exponential modes.
    The damping of the short-period mode improves with an increase in the stable
values of Cmq and Cma,- Recall that the major contr:ibution to Cmq and Cma comes
from the horizontal tail. The higher the tail-volume ratio, the larger will be the
horizontal tail contribution to Cmq and Cma and the higher will be the short-period
damping ratio.
6.2.2 PhugoidApproximation
   Because the disturbance in angle of attack quickly decays to zero during the
short-period oscillation and subsequently remains close to zero, we assume Act -
Aa  = O for the phugoid motion. Furthermore, we assume that the pjtching motion
is quite slow so that pitch acceleration can be ignored, i.e., q = 0 - 0. In view
of this, the pitching moment Eq. (6.3) can be ignored. Then, Eqs. (6.1) and (6.2)
548            PERFORMANCE, STABILfTY, DYNAMICS, AND CONTROL
reduce to the following form:
                   mi ddt = Cxuu + Cx0AO + Cxq.,(dC~O) + Cx8t A8e         (6.58)
      .dAO
                  (mi + CzqCI) dt  = -Cz"u - CzOAO - Cz8.A8e          (6.59)
Rearranging in a form suitable for state-space representation, we obtain
Let
dd =(
so that
where
Cxu  - €3 Czu
mi
).+(
Cx0 - 93Cz0
mi
)^0+(
                  Cxq Cl
g3 - =-
            TTZ1 + Czq CI
XI -U
 X2 -. AO
X = ~:]
 U - A8e
CX3e - g3C23e
mi
)A8e
(6.60)
)A8, (6.61)
(6.62)
(6.63)
(6.64)
(6.65)
(6.66)
[x"::] = ~,"'   /-2] [::] + [::] U                   (6.67)
Aph = K"'  /~1
Bph =
aii=(C g_C)
a2,=( +C., C,,.,)
b,=(C g_C)
[::]
           Cx0 - 93
a12- -  g-C )
     mi
  -Czo
a22 ~  -
                 Ftll + Czq Cl
b2- -Cz8e
                  tTIl + Czq C\
(6.68)
(6.69)
AIRPLANE RESPONSE AND CLOSED-LOOP CONTROL         549
Equations (6.60) and (6.61) are approximate equations for the phugoid or long-
period mode and are applicable-for conventional statically stable airplanes. These
equations may not be applicable if the airplane is statically unstable or incorpo-
rates relaxed static stability concepts wherein the aircraft is rendered statically
unstable to improve the performance. For such airplanes, as we will discuss later,
the conventional short-period and phugoid modes do not exist.
   To get an idea of the physical parameters that have major influence on the
frequency and damping ofthe phugoid mode, we need to introduce some additional
simplifications. As said before, Cz0 ~ 0. Furthermore, assume that Ciq and Czq
are smaU and can be ignored. With these assumptions, it can be shown that the
　
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