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self. Because the period is quite long, the pilot can easily operate the longitudinal
control (elevator) and kill the phugoid mode.
Physically, the motion of the airplane during the phugoid motion can be depicted
as shown in Fig. 6.3. Beginning at the bottom ofone cycle, we observe that the pitch
angle increases and the airplane gains altitude and loses forward speed. During
phugoid motion, the angle of attack remains constant so that a drop in forward
speed amounts to a decrease in lift and flattening of the pitch attitude. As a restrlt,
at the.top of the cycle, the pitch angle goes to zero. Beyond this point, the airplane
begins to lose altitude, the pitch angle goes negative, and the air speed increases.
At the bottom of the cycle, the pitch attitude is nearly level, the air speed is at its
maximum, and the cycle repeats once again.
AIRPI_ANE RESPONSE AND CLOSED-LOOP CONTROL ' 545
Fig. 63 Schematic illustration of the physical motion of the airplane during the
phugoid moOon.
6.2.I Short-PeriodApproximation
Ofthe two oscillatory modes, the short-period mode is the more heavily damped
oscillator}r motion with a higher frequency. This oscillator}r motion lasts just for
a few seconds, usually fewer than 10 s, during which the angle of attack, pitch
angle, and pitch rate vary rapidly and ffie forward speed nearly remaints constant.
Therefore, we can assume u : ri : O during short-period oscillation. With this
assumption, we can ignore the force equation in the x direction. Then, the other
two equations assume the following form:
dd_
m] dt - Cz&cl dt - Cz.) \ _ (m,dd
-(C...,~+C. )Aa+dd(.ty,dd -C"q.,)AO=Cm8eA8e (6.42)
Let
Then rearranging, we have
XI - Act
X2 -q
X3 - AO
(6.43)
(6.44)
(6.45)
x::l=[a:'~ ai: aj:l[x:'l+[b:'lA8e (6.46)
In view of the fact that the longitudinal response of statically stable airplanes
consists of two distinct oscillator)r motions, it is customary to introduce the short-
period and phugoid approximations as follows.
+ Czq C, 3t + CzO)AO = Cz,e A8e
(6.41)
546 PERFORMANCE, STABfLITY, DYNAMICS, AND CONTROL
where
We have
Asp = [a2"'
a12 a13]
022 a23
a32 a33j
Bsp = [b;:]
Cza
aii ::= -
mi' z ci
a13= Cz0
mi - Czd,CI
rt? i + Czq Cl
a12 - -
mi - Czd,CI
a2,=(} (C,.+ C.
mi - Czdt CI
a22=(11) CmqC,+( C C )(,,+CICzq)]
C,n *ci Cz0
az:4 - I7i(mi - CzdCI) a31 =0
bi = _Cz8.
mi - Cz&ci
a32 -.1 033 - O
b2= 1)(C,&+C."-CC,atl
b3 =0
Cz0 - -CL siri Oo
For equilibrium level flight, 0o = O, so that Cz0 ~ O, a13 -
this assumption, Eq. (6.46) reduces to the following form:
[;~:] = [:;:: t;22] [::] + [::] A8e
(6.47)
(6.48)
. (6.49)
0, and a23 -. O. With
(6.50)
To get an idea of the physical, parameters that have a major influence on the
short-period mode, we have to introduce some more simplifications. We assume
Czq = CzCr - O because they are usually small. With these assumptions,it can be
shown that the characteristic equation of the short-period mode is given by
A2+BA+C _ 0
(6.5 1)
where
AIRPLANE RESPONSE AND CLOSED-LOOP CONTROL 547
B=-(C + j, (Cmq+C,,))
C~CzaCICmq Cma
m i 11.1 11.1
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