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REV/EW OF BAS\C AERODYNAMIC PRINCIPLES
Fig.1.20 Schematici/lustrationofairfoilcharacteristics'
20 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
the moment reference point. The sign of g depends on the location ofthe moment
reference point. If this point is located close to the leading edge, the incrementallift
caused by angle ofattack acts aft of the moment reference point, and the associated
incrementin pitclung moment will be negative, making g < 0. On the other hand,if
the moment reference pointislocated close to the trailing edge, then theincremental
lift acts ahead of the moment reference point, and the corresponding incremental
pitching moment is positive; hence g > 0. Therefore, there must be some point
on the chordline where g = 0 so that the incremental pitcfung moment caused
by angle of attack is zero and the pitching-moment coefficient remains constant
with respect to angle of attack and equal to Cmo. This point is called aerodynamic
center. In other words, all the incremental lift caused by angle of attack acts at the
aerodynamic center. The pitching-moment coefficient with aerodynamic center as
the moment reference point is denoted by C . Note that
Cmac - Cmo
(1.35)
For symmetric airfoils, Cmac - Cmn - O.
The concept of aerodynamic center is helpful in the study of airplane stability
and control because it is sufficient to specify the pitching-moment coefficient at
any one angle of attack'in the entire linear range of angle of attack.
At low speeds, the aerodynamic center usually lies close to the chordline and is
located at approximately 22 to 26% chord f:rom the leading edge. For approximate
purposes,it can be assumed that it is located on the chordline at the quarter chord
point. For high subsonic and supersoruc speeds, both the aerodynamic center and
center of pressure move aft.
The exact location of the aerodynamic center can be determined if given the
lift, drag, and pitching-moment coefficients about any reference point along the
chordline. Let O be the moment reference point (see Fig: 1.21a) about which
we are given q, Cd, and Cm. Let Xe,c be the distance of the aerodynamic center
from O, positive aft. Let Cmac denote the pitching moment coefficient about the
aerodynamic center, which by definition is invariant with angle of attack. Then,
the pitching moment about the given moment reference point can be written as
In terms of coefficients
M - Mac - xucL
Cm - Cmac - XacCl
(1.36)
(1.37)
where Xac - Xac/C. Differentiate with respect to Ci and note that Cmac (by defini-
tion) is constant. Then
or
(1Cm _
dCt = -xuc
- dCm
Xcrc = - dCi
(1.38)
(1.39)
Thus, forlinear range of attack, x"c depends on the slope ofthe pitching moment
curve.IfdCm ~dCt > 0, then xoc is negative,implying that the aerodynamic centeris
REVIEW OF BASIC AERODYNAMIC PRINCtPLES 21
Vm
a)
Aerodynamic Centcr
flg.l 21 Concept of aerodynamic center and center ofpressure.
1-5.2 Relation Between Centerof Pressureand Aerodynamic Center
For linear range of angle of attack,
Cm = Cmo + ddqC C,
(1.40)
' = -XcpCI (1.41)
where Xcp = Xcp/C and Cmo iS the pitching-moment coefficient at that angle of
attack when the lift-coefficient is zero. Let tYOL denote this angle of attack. As said
before, for symmetric airfoil sections, OCOL = O; for sections with positive camber,
aOL is negative; and for sections with negative camber, crOL is positive.
Then,
- Cmo dCm
Xcp - --
- Cl - dCi
(1.42)
22 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Fig. 1.22 Variation of center of pressure Iocahon with angle of attack.
Using Eq. (1.39), we get
. Cmo
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