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angles of attacktsideslip when the aerodynanucvcoefficients vary linearly with an-
gle of attack/sideslip. At high angles of attack/sideslip, such assumptions are not
valid because offlow separat:ion, vortex shedding, and stall. As a result, the aerody-
namic coefficients vaxy nonlinearly with angle of attack/sideslip, and aerodynamic
coupling takes place. When this happens, a change in the angle of attack affects
side force, rolling, and yawing moments. Similarly, a change in sideslip angle
influences lift, drag, and pitching moments. We will study stability and control
problems at high angles of attack in Chapter 8.
With these assumptions and remember:ing that the disturbance variables are
assumed to be small, we can use the Taylor series expansion method around the
equilibrium level flight condition to obtain the forces and moments in the disturbed
state as follows: .
AC, = aac, .+ aa~Act + aacZAO + aa~A& + aaC^q q
ac
+ aa~- A8e + aa~ A8,+ - -
. acy
ACy = aC~A6+ aacj A4+ aaclp AB + aacYp P
+ aa~r + aa~A8" + aa~A8r +..-
t a(
ACz = aa~u + aa~Aa + aa~A8 + aa~A& + aaCg~q
a
. +aa~A8.+aa~A8,+--
AC, = aa~Ap + aack Ap + aac~A* + aa~P
+ aa~r + ~~-A8a + aa~A8r + *..
(4.399)
(4.400)
(4.401)
(4.402)
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 377
acm
ACm = aa~u+ aac Aa+ aac"o AO+ aa~Ad+ 8aq q
ac
+ aa~-A8e + aa~z\8,+... (4.403)
ACn = aac~Ap + aac~Ap+ aac4 ~+ aaC;.p
+ aa~r + aa~A8a + aa~A8r +... (4.404)
In Eqs. (4.399-4.404), 8e is the elevator/elevon deflection, 8a is the aileron deflec-
tion, 8r is the rudder defiection, and 8r is the engine control parameter.
Terr~is such as aCc/a u, a cz/au, ~d a cmlaU &e stability derivatives, and terms
such ~ a cm~a8e md a Cm ]88t are control derivatives.lt should be noted that these
stability and control der:ivatives are evaluated at the equilibrium flight condition
from which the airplane is supposed to be disturbed. T7:erefore,it is possible that
stability and control derivatives vary from one equilibrium condition to another.
It is important to remember that the Taylor series expansion must include all
the motion variables on which the aerodynamic forces and moments are known
to depend. If this information is incomplete, estimated forces and moments in
the disturbed state will be incorrect. Any predictions based on such incomplete
aerodynamic data will also bein error. Therefore,itis the task ofthe aerodynamicist
to understand the physics ofthe problem,identify all the motion variables on wluch
the aerodynamic forces and moments show dependence, and correctly include all
of them in the Taylor series expansion.
In the literature on airplane stability and control, it is customar)r to use the
short-hand notation to denote the stability and control derivatives. For example,
, aq acx acm
.x"=a~ cxu=a Cmcr= (4.405)
au : ay = a-
acn
Cyp=88~ cl,,=aa2 Cp=ap (4.406)
and so on. However, the partial derivatives with respect to variables such as
d, B, p, q, and r are deftned somewhat differently as follows:
C,d = aa(:c:.)
CyB = (, ~)
Cyp= a(:c..)
Cyr= a~c~
C q = aa(:c: )
CiB = (, A~,.,)
Clp = (, ::,)
Cmq = aa(c,:.)
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PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL2(130)
