User:Eas4200c.f08.wiki.f/10-6-08

Quadrature Method
Quadratures can be used to find the average area of foils. Below is an image of the quadrature of a foil.



$$\overline{A} = \overline{A}_1 + \overline{A}_2 = \overline{A}_1 - \begin{vmatrix}\overline{A}_2\end{vmatrix}$$
 * $$ \overline{A}_1 > 0 $$
 * $$ \overline{A}_2 > 0 $$




 * $$\overrightarrow{r}x\overrightarrow{PQ}\; goes\; in\; negative\; x\; direction\; $$

Using Trapezoidals to Integrate
There are problems with using this method because of the curvature of the foil. When you split up the area of the foil, not every section will end up looking like a trapezoid. This method should not be used for foils.



Shear Flow in a Single Cell Airfoil
Below is an image of a single cell airfoil.



The shear flow is constant, therefore:


 * $$ q=q_1=q_2=q_3$$

Rate of Twist Angle θ

$$\theta = \frac{1}{2G\overline{A}}\; q\sum_{j=1}^{}{\frac{l_j}{t_j}}$$

$$\theta = \frac{1}{2G\overline{A}}\; q[\frac{\pi (\frac{b}{2})}{t_1} + \frac{a}{t_2} + \frac{\sqrt{a^2 + b^2}}{t_2}]$$

$$\theta = \frac{1}{2G\overline{A}}\; q[\frac{\pi bt_2t_3 + 2at_1t_3 + 2t_1t_2\sqrt{a^2 + b^2}}{t_1t_2t_3}]$$

Shear Stress
Max Shear Stress τmax

$$\tau_{max} = \frac{q}{min(t_1, t_2, t_3)}$$


 * $$if\; \tau_{max} = \tau_{all}$$


 * $$and\; since\; q = \frac{T}{2\overline{A}} \; then$$

$$T_{all} = 2\bar{A}\tau_{all}\;min(t_1, t_2, t_3)$$