Evaluating double integrals for bending-torsion coupling constants

[zmall88]

I've been trying to compute the bending-torsion coupling constants for a wing, B₁, B₂ and B₃. The expression for this is
$\begin{bmatrix} B_1 \\ B_2 \\ B_3 \end{bmatrix} = \iint (y^2 + z^2)\begin{bmatrix} y^2 + z^2 \\ z \\ y \end{bmatrix}dydyz$
where x is in along the wingspan direction, y is along chordwise direction and z is perp. to both.

Question is: how to evaluate this integral?

I have z as a series of points (airfoil shape), where at every y, there are two z values (upper and lower surfaces).

I'm not sure this is in the correct forum or not...

 

[Navin A S]

but I'm hoping someone can help me out.The integral you have written is a double integral and it can be evaluated by breaking it up into two single integrals. Specifically, you can first integrate over y and then integrate over z. To do this, you need to know the limits of your integration. For example, if the wing has a span of 10m, the limits of your integration in the x direction would be 0 to 10m. Similarly, you need to know the limits of your integration in the y and z directions. Once you have set the limits of integration, you can break up the double integral into two single integrals:\begin{align}B_1 &= \int_{y_{min}}^{y_{max}} \left(\int_{z_{min}}^{z_{max}} (y^2 + z^2)(y^2 + z^2)dz\right)dy \\B_2 &= \int_{y_{min}}^{y_{max}} \left(\int_{z_{min}}^{z_{max}} (y^2 + z^2)zdz\right)dy \\B_3 &= \int_{y_{min}}^{y_{max}} \left(\int_{z_{min}}^{z_{max}} (y^2 + z^2)ydz\right)dy\end{align}These single integrals can then be evaluated using standard numerical methods.