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tana ~ Aa - -
Uo
sinp ~ Ap = jt
376 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
4.3.5 Estimation of Aerodynamic Forces and Moments
Our next task is to estimate the aerodynamic coefficients ACx, ACz, ACm,
A/, AM, and AN in the disturbed motion. For this purpose, we use Bryan's
method,i which, as said- in section 4.1, is based on two principal assumptions:
1) the instantaneous aerodynamic forces and moments depend on the instaneous
values of the motion variables and 2) the aerodynamic forces and moments varjr
linearly with motion variables. Furthermore, we introduce another important as-
sumption that the longitudinal aerodynamic forces and moment (Fx,'$~ and M)
are infiuenced only by the longitudinal variables u, a, and q. In other words, we
assume that the lateral- directional variables p, p, and r do not infiuence the lon-
gitudinal aerodynamic forces Fx and Fz and the pitching moment M. Similarly,
we assume that the side force Fy, the rolling moment L, and the yawing mo-
ment N depend only on the lateral-directional variables p, p, and r and are not
influenced by the longitudinal variables u, a, and q. In other words, we assume
that there is no aerodynamic coupling between longitudinal and lateral-directional
variables, forces, and moments. These assumptions are usually valid for small
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
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