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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
X,p = xac -  Cf
(1,43)
     Thus, the location of the aerodynamic center relative to the center of pressure
depends on the sign of Cmo. For positively cambered'aufoils, Cmo <0 SO that
the center of pressure is aft of the~erodynamic center as shown in Fig. 1.21b.
However, as the angle of attack increases (O s Cl s Cl.max), the center of pressure
moves towards the aerodynanuc center as shown in Fig. 1.22.
1.5.3  Stallof Wing Sections
        Atlow angles of attack, the flow separation occurs at or close to the trailing edge.
As the angle of attackincreases, the separation point gradually moves towards the
leading edge.ln this process, the lift coefficient continues to increase and, at some
point, attains a maximum value. Beyond this value of angle of attack, the lift
coefficient drops, and the airfoilis said to have stalled. This type of stall generally
occurs on thick airfoils and is characterized by a gradualloss of lift beyond the
stall as shown by the curve a in Fig.  1.23.
     For thin airfoils that have sharply curved leading edges, a slightly different type
of stall occurs. From the leading edge and up to the minimum pressure point, a
　
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