User:Eas4200c.f08.blue.a/Lecture 17



Cubature - a method used for integrating volumes by breaking up the object into cubes and adding the individual cubes to find the total volume. To make the method more accurate, use smaller cubes to better approximate the volume.

Squaring (quadrature) of the circle - wikipedia.



Algebraic sums of the areas as the airfoil is swept counterclockwise is the NACA airfoil area.



A = A1 + A2 = A1 - |A2|

NOTE: When writing the code plotting the NACA arifoil, we do not need if statements to calculate the area if the arifoil. As long as the cross product is taken between the 2 sweeping vectors, the signs will add up correctly to give the appropriate area.



How about using trapezoids to integrate??



The method of trapezoids is easier to integrate over the surface and gives a more precise answer than Riemann Sums however it has a distinct disadvantage over the method of quadrature. The disadvantage being, changes in curvature are not accurately taken into account. For example, a segment with a lot of curvature is not approximated very well with a straight line. By using the method of quadrature, as long as there are a large number of segments, the area can be approximated better. Not to mention, THE TRAPEZOID METHOD IS NOT AS ELEGANT AS THE METHOD OF QUADRATURE (TRIANGLES)!!!!!!!

Returning to the single cell airfoil:

Shear flow along the airfoil is constant: q = q1 = q2 = q3

The angular rate of twist is:

$$ \theta =\frac{1}{2GA}q\sum_{j=1}^3 \frac{l}{t} $$

$$ \theta =\frac{1}{2GA}q [\frac{\pi b}{t1} + \frac{a}{t2} + \frac{\sqrt{a^2 + b^2}}{t3}] $$

$$ \theta =(factor)q $$

Homework: Computing the factor for the rate of twist.

Max shear stress = tau_max

If tau_max = tau_all (which is given), and since q = T/2A then,

Tall = 2Atauall[min{t1,t2,t3}]

Homework: If tau_all = 100 Gpa, fint T_all.