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characteristic equation of the phugoid mode is given by
A2 _ Cxrr A+ C o4- = 0
m i lTI21
(6.70)
Comparing this equation with that of the standard second-order system shown as
Eq. (6.34j5we obtain
1
tOn = ~
m
Cxu
c = -2rn IWn
We have
C^u - -2CD - CDu
Cx0 ~ -CL COS 0o
Cz" - -2CL - CLu
(6.71)
(6.72)
(6.73)
(6.74)
(6.75)
For low subsonic speeds, CD ~ ~ CLu ~ O. For level fiight, 0o - 0 so that
Cxu - -2CD (6.76)
Cx0 - -CL (6.77
Czu = -2CL (6.78)
Furthermore, using CL - 2 WlpUo2S, we obtain
tOn = ~g
T = .~7r Ut,
g
g=~(g )
1
= EE
(6.79)
(6.80)
(6.81)
(6.82)
dAO -Czu -Cz8t
d =( +C., ,,,,).+( +C., C,,,,)AO+( +~CzqCI
where
550 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Thus, we observe that the damping of the phugoid mode isinversely proportional to
the aerodynamic efficiency E. Because E varies with angle of attacK the phugoid
damping will vary with angle of attack, becoming minimum when the airplane
flies at that angle of attack when E - Emax or CL = J iF. We ~so note that
the period of the phugoid motion increases with forward speed. \
6.2.3 Accuracy of Short-Period and Phugoid Approximations
It is instructive to assess the accuracy of the short-period and the phugoid ap-
proximations. For this purpose, let us consider the general aviation airplane once
again as follows.
Substituting the values of mass, inertia, and aerodynamic parameters in Eqs.
(6.46) and (6.67), we get
-2.0354 0.9723 1
Asp= -7.0301 -2.9767j
[-.::l~;i]
Aph = [ 003~42533 0003138g]
. Bph = [0.0~29]
(6.83)
(6.84)
(6.85)
(6.86)
Using MATLAB,i we obtain the eigenvalues for the short-period and phugoid
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