Page 261 - Aerodynamics for Engineering Students

244 Aerodynamics for Engineering Students
Now in addition the local chord can be expressed as a fraction of the semi-span s, and
with this fraction absorbed in a new number and the numeral 4 introduced for later
convenience, I? becomes:
r = 4crs
where Cr is dimensionless circulation which will vary similarly to r across the span.
In other words, Cr is the shape parameter or variation of the I' curve and being
dimensionless it can be expressed as the Fourier sine series ETA, sin ne in which the
coefficients A,, represent the amplitudes, and the sum of the successive harmonics
describes the shape. The sine series was chosen to satisfy the end conditions of the
curve reducing to zero at the tips where y = As. These correspond to the values of
0 = 0 and R. It is well understood that such a series is unlimited in angular measure
but the portions beyond 0 and n can be disregarded here. Further, the series can fit
any shape of curve but, in general, for rapidly changing distributions as shown by
a rugged curve, for example, many harmonics are required to produce a sum that is
a good representation.
In particular the series is simplified for the symmetrical loading case when the even
terms disappear (Fig. 5.32 01)). For the symmetrical case a maximum or minimum
must appear at the mid-section. This is only possible for sines of odd values of 742..
That is, the symmetrical loading must be the sum of symmetrical harmonics. Odd

I

0 x 7r
2
-S 0 S

Fig. 5.32 Loading make-up by selected sine series