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AC,n : Cm,,u + Cmr Aa + C,,,o AO + C, , (~U ) + C"q (2qU )
+ Cm8e A8e + Cm& A8r + . . . (4.415)
ACn = Cnp A8 + C.B (jljb) + Cn~A~ + Cnp (2pUb ) + Cnr (2;;.)
+ Cnaa A8a + C,,tr Ab'r + . - . (4.416)
Itisinteresting to observe that, with the assumption of small distwrbances andlin-
ear uncoupled aerodynamics, Eqs. (4.393), (4.395), and (4.397) with Eqs. (4.411),
(4.413), and (4.415) contain only the longitudinal disturbance variables and lon-
gitudinal aerodynamic forces and moment. Similarly, Eqs. (4.394), (4.396), and
(4.398) with Eqs. (4.412), (4.414), and (4.416) contain only the lateral-directional
disturbance variables and lateral-directional aerodynamic force and moments.
Thus, we can separate the longitudinal and lateral-directional small disturbance
equations of motion into two different sets and solve them separately. In this
way, we have reduced a coupled nonlinear six-degree-of~freedom problem to two
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 379
separate problems of three degrees of freedom, both of which are linear with re~
spect to the disturbance variables. The linearity of the equations makes it easy
for us to apply the powerful analytical methodsYoflinear 9ontrol systems for the
study of airplane dynamic stability, control, and response calculations. However,
it is necessary to keep in mind that this simplification is based on small distur-
bances and linear aerodynamics. Obviously, thP;S approach is not valid for complex
airplane maneu'vers involving large amplitude motion typically occurring at high
angles attack and stall/spin.
Longitudina/ equations of motion for sma// disturbance. Substituting for
ACx, ACz, and ACm from Eqs. (4.411), (4.413), and (4.415) in Eqs. (4.393),
(4.395), and (4.397), we obtain
d d
1T11(1, - Cx,.). - (Cxd C, dd + Cxa) A, - (Cvq Cl d + CxO)AO
- Cx8e A8e + Cxbr A8t (4.417)
-Czuu + (m,dd _ Czcrrci dd ) - Cz,,]Aa _ (m,dd + CzqCl dd + Czo)AO
= CZSe A8e + Cz8t A8t . (4.418)
-C, ,. - (Cm*ci dd + Cm.) Aa- + dd (/y, dd _ Cmq,,) AO
= CmlSe A8e + Cm8t A8t
- (4.419)
where
c
Cl = 2U (4.420)
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