Page 270 - Aerodynamics for Engineering Students

Finite wing theory 253

Table 5.1

7~/8 0.382 68 0.923 88 0.923 88 0.382 68 0.923 88
~14 0.707 11 0.707 11 -0.707 11 -0.707 11 0.707 11
3~18 0.923 88 -0.382 68 -0.38268 0.923 88 0.38268
7512 1 .ooo 00 - 1 .ooo 00 1 .ooo 00 - 1 .ooo 00 0.000 00

This gives at any section:

and
par = 0.032995(i+o.5cOse)(i - o.o5455~0se)(i +0.36364cosq

where a! is now in radians. For convenience Eqn (5.60) is rearranged to:

par sinB=AlsinO(sin8+p) +A3sin3f3(sin8+3p) +A5sin50(sinO+5p)
+ A7 sin 78(sin 8 + 7p)

and since the distribution is symmetrical the odd coefficients only will appear. Four coefficients
will be evaluated and because of symmetry it is only necessary to take values of 8 between 0 and
~12, Le. n-18, n/4, 3~18, 42.
Table 5.1 gives values of sin 0, sin ne, and cos 8 for the above angles and these substituted in
the rearranged Eqn (5.60) lead to the following four simultaneous equations in the unknown
coefficients.

0.004739 = 0.22079 A1 + 0.89202 A3 + 1.251 00 A5 + 0.66688 A7
0.011637 = 0.663 19 A1 f0.98957 A3 - 1.315 95A5 - 1.64234 A7
0.0216 65 = 1.1 15 73 A1 - 0.679 35 A3 - 0.896 54 A5 + 2.688 78 A7
0.032998 = 1.343 75 AI - 2.031 25 A3 - 2.718 75 A5 - 3.40625 A7
These equations when solved give

A1 = 0.020 329, A3 = -0.000 955; A5 = 0.001 029; A7 = -0.000 2766
Thus

r = 4sY{0.020 329 sin 8 - 0.000 955 sin 38 + 0.001 029 sin 50 - 0.000 2766 sin 78)
and substituting the values of 8 taken above, the circulation takes the values of:

4s 1 0.924 0.707 0.383 0
rm2s-I 0 16.85 28.7 40.2 49.2
Firo 0 0.343 0.383 0.82 1 .o