Page 207 - Aerodynamics for Engineering Students
P. 207
190 Aerodynamics for Engineering Students
For the theoretical estimation of zq and m4, of the complete aircraft, the contribu-
tions of the tailplane must be added. These are given here for completeness.
(4.75)
where the terms with dashes refer to tailplane data.
P I .
4.8 Particular camber lines
It has been shown that quite general camber lines may be used in the theory
satisfactorily and reasonable predictions of the aerofoil characteristics obtained.
The reverse problem may be of more interest to the aerofoil designer who wishes to
obtain the camber-line shape to produce certain desirable characteristics. The general
design problem is more comprehensive than this simple statement suggests and the
theory so far dealt with is capable of considerable extension involving the introduc-
tion of thickness functions to give shape to the camber line. This is outlined in
Section 4.9.
4.8.1 Cubic camber lines
Starting with a desirable aerodynamic characteristic the simpler problem will be
considered here. Numerous authorities* have taken a cubic equation as the general
shape and evaluated the coefficients required to give the aerofoil the characteristic of
a fixed centre of pressure. The resulting camber line has the reflex trailing edge which
is the well-known feature of this characteristic.
Example 4.1 Find the cubic camber line that will provide zero pitching moment about the
quarter chord point for a given camber.
The general equation for a cubic can be written as y = a’x(x + h’)(x + d’) with the origin at the
leading edge. For convenience the new variables x1 = x/c and 1’1 = p/b can be introduced. b is
the camber. The conditions to be satisfied are that:
(i) .y = 0 when x = 0, Le. yl = XI = 0 at leading edge
(ii) y = 0 when x = c, i.e. yl = 0 when XI = 1
(iii) dy/dx = 0 and y = 6, Le. dyl/dxl = 0 when yl = 1 (when x1 = .x”)
(iv) CM, = 0, i.e. A1 - A2 = 0
Rewriting the cubic in the dimensionless variables xI and yr
y1 = ax1 (XI + b)(x1 + d) (4.76)
this satisfies condition (i).
To satisfy condition (ii), (XI + d) = 0 when XI = 1, therefore d = -1, giving
1’1 = axi(x] +h)(Xl - 1) (4.77)
or multiplying out
yl = ux; + u(b - 1)x; ~ uhxl (4.7X)
* H. Glauert, Aerqfoil and Airscrew Theory; N.A.V. Piercy, Aeroclynumic,~; etc.
