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damped oscillatory motion if the real part of the complex root is negative, and a
divergent oscillatory motion if the real part of the complex root is positive. rfhus
for dynamic sta'oility, the roots of the characteristic equation must be negative if
real or must have negative real parts if complex.
For a conventional, staticallyPs"rjable airplane (Cma < O), the longitudinal char-
acteristic equation usually has a pair of complex conjugate roots of the form
A1.2 = -ri +_ jsi
A3.4 -. -r2 :1: js2
(6.26)
(6.27)
542 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
NOMINAL FLIGHT CONOITION
iYN~r
h{t.) i0 . r -.is8 i vo -l?6 t.Nc
w - Z73O lb
C6 ot 29.l~X MAC
lI . l048 dW ttt
Iy , 3000slu0 tti
rll . H30 rluq tti
Iu 'O
ReFEfIENCe 6EWJETRY
_
s - l84 ttt
c - 5.7 tt
b i 33.4 tt
so that
Fig. 6.2 rfhree_view drawing of the general aviation airplane.z
Along(A) = (-ri + jsiX-ri - jsl)(-r2 + js2)(-r2 - js2) (6.28)
The roots of the characteristic equation are also callecl the eigenvalues of the
system represented by the matrix A. To determine the eigenvalues A/, / ~ 1, 4, we
can either solve the fourth-order characteristic equation or obtain them directly
using the methods of matrix algebra. However. either of these tasks can be easily
performed using commercially available matrix software tools like MATLAB.1
To illustrate the nature of free response and the concept of airplane dynamic sta-
bility, let us determine the response of a general aviation airplane,2 which is shown
in Fig. 6.2. The mass and aerodynamic properties of this airplane are as follows.2
V,hng area S = 16.7225 m2, weight W = 12,232.6 N, wing span b = 10.1803 m,
wing mean aerodynamic chord c - 1.7374 m, distance of center of gravity from
wing leading edge in terms of mean aerodynamic chorc -0.295, Ix -
1420.8973 kg/mz, I). = 4067.454 kg/m2, Iz = 4786.0375 kg/dm :9and Ixy = Iyz =
Izx - 0. CL -.0.41, CD = 0-05, Cl.a - 4.44, CIA, - O, CDtr = O, CLM - O, CL3e -
0.355, Cm3e = - 0.9230, CDu - 0.33, CDNt = 0, CD8e = 0, Cma = - 0.683, CmCr -
-4.36, CmM = O, Cmq = -9.96, M = 0.158, and p = 1.225 kg/m3. All the deriva-
tives are per'radian.
We assume that the airplane is in level flight with Qo - 0 before it encounters
any disturbance. This assumption gives us Cx0 - -CL and Cz0 -. O.
AIRPLANE RESPONSE AND CLOSED-LOOP CONTROL 543
Substituting the required quantities in Eq. (6.12), we get
-0.0453
A-. -0.3717
0.3398
O
0.0363
-2.0354
-7.0301
0
0
0.9723
-2.9767
1
Using MATLAB,I we obtain the eigenvalues of the matrix A as
A1.2 = -2.5227 + j2.5758
A3.4 = -0.0060:1: j0.2133
(6.29)
(6.30)
(6.31)
(6.32)
Because both the complex roots have negative real parts, the free response of this
airplane is of stable nature. It consists of t.wo decaying oscillatory motions, which
are superposed one on another. Thus, the general aviation airplane considered here
is dynamically stable.
An alternative way of examining the dynamic stability of the airplane without
actually solving for the roots of the characteristic polynomial or the eigenvalues
of the matrix A is to make use of the Routh's stability criterion we discussed in
Chapter -5. Substituting in Eq. (6.20), the characteristic equation for the general
aviation airplane is given by
0.3739 A4 + 1.9002 A,3 + 4.9935 A,2 + 0.1642 A, + 0.2296 = 0 (6.33)
The necessary condition for stability is that all the coefficients of the characteristic
equation must be positive or have the same sign. This condition is satisfied here.
To examine whether the sufficiency condition is satisfied, we form Routh's array
as follows:
S4 : 0.3739 4.9935 0.2296
S3 : 1.9002 0.1642 0
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