Mean Aerodynamic Chord, Mean Aerodynamic Center, and Cm
C. l Basic Definitions
In the normal flight range, the resultant aerodynamic forces acting on any lifting surface can be represented as a lift and drag acting at the mean aerodynamic center (x, y, z), together with a pitching couple C„,ut which is independent of angle of attack (see Fig. 2.8).
The pitching moment of a wing is nondimensionalized by the use of the mean aerodynamic chord c.
Both the m. a. center and the m. a. chord lie in the plane of symmetry of the wing. However, in determining them it is convenient to work with the half-wing.
These quantities are defined by (see Fig. C. l)1
c=- C2dy (C.1,1)
2 (Ы 2
x==-— C, cxdy (C-1,2)
2 rw2 b
У = 777 I C, cydy = Пер – z (C-1,3)
CLS Jo I
Z=yzz Ctczdy (C.1,4)
where b = wing span
c = local chord CL = total lift coefficient
Ci = local additional lift coefficient, proportional to CL Clh = local basic lift coefficient, independent of C,
C, = C, b + C, a = total local lift coefficient
‘The coordinate system used applies only to this appendix.
mac = pitching moment, per unit span, about aerodynamic center (Fig. C.4)
S = wing area
у = spanwise coordinate of local aerodynamic center measured from axis of symmetry
x = chordwise coordinate of local aerodynamic center measured aft of wing apex
г = vertical coordinate of local aerodynamic center measured from xy plane VcP = lateral position of the center of pressure of the additional load on the halfwing as a fraction of the semispan
The coordinates of the m. a. center depend on the additional load distribution; hence the position of the true m. a. center will vary with wing angle of attack if the form of the additional loading varies with angle of attack. For a wing that has no aerodynamic twist, the m. a. center of the half-wing is also the center of pressure of the half-wing. If there is a basic loading (i. e., at zero overall lift, due to wing twist), then (x, y, z) is the center of pressure of the additional loading.
The height and spanwise position of the local aerodynamic centers may be assumed known, and hence у and z for the half-wing can be calculated once the additional spanwise loading distribution is known. However, in order to calculate x, the fore-and-aft position of each local aerodynamic center must be known first. If all the local aerodynamic centers are assumed to lie on the nth-chord line (assumed to be straight), then
x = ncr + у tan A„
where cr = wing root chord
An = sweepback of nth-chord line, degrees
Ideal two-dimensional flow theory gives n = for subsonic speeds and n = | for supersonic speeds.
The m. a. chord is located relative to the wing by the following procedure:
1. In (С. 1,2) replace Cla by CL, and for x use the coordinates of the і-chord line.
2. The value of x so obtained (the mean quarter-chord point) is the j-point of the m. a. chord.
The above procedure and the definition of c (see С. 1,1) are used for all wings.