I was reading Wright and Cooper's writings on static aeroelasticity and I think I'm missing something. To calculate the acting lift, the equation $L = q S a_1 \theta_0$ is used, where $q$ is the dynamic pressure, $S$ is the wing area, $a_1$ is the lift-curve slope ($\frac{\delta C_L}{\delta \alpha}$) and $\theta_0$ is the initial incidence angle.

However, the way I see it, that only equals to the traditional $L = qS C_L$ when/if I have a curve where both $C_L$ and $\alpha$ are zero at the same time, since the slope indicates the rate of my $C_L$ variation but not the starting point.

So the question is: is the Wright and Cooper's formula only supposed to calculate the increase in lift generated or am I reading it wrong?

I'll try and show a few examples of where that is used, so it may clarify.

In divergence studies, to calculate the total lift:

$L_{total} = L_{rigid} + L_{elastic} = q S \frac{\delta C_L}{\delta \alpha} (\alpha_0 + \theta)$

Where $\alpha_0$ is the original angle of attack and $\theta$ is the twist angle due to aerodynamic loading. The way I see it, this should be equal to $\large L_{total} = qS C_{L_{\alpha+\theta}}$, but it isn't. I'm not sure if it's not supposed to, or if I'm seeing it wrong.

In Wright and Cooper's book, to calculate the pitching moment on an airfoil.

Example from section 8.1.1 in Wright and Cooper's Introduction to Aircraft Aeroelasticity Two dimensional aerofoil with a torsional spring, as described by W&C

Here again, they calculate lift as $L = q c a_1 \theta_0$, and that is not equal to $L = qc C_L$, because the admitted $C_L$ values will be lower.

I hope I made myself sufficiently clear, thank you in advance.

Edit: added figure 8.1 and tried to clarify a bit.

  • $\begingroup$ Hi welcome to Engineering. Could you upload an image of 8.1? Also could you sum up your question? Are you considering the general case or the example in the book? What is that you are expecting? What is the difference to what your reference is saying? $\endgroup$
    – NMech
    Apr 19 '21 at 22:35
  • $\begingroup$ @NMech Hi, thank you. I will add the figure 8.1 to my original post, if that's what you're asking. The question is why Wright and Cooper describe lift that way if it leads to incorrect values. I understand he's not wrong, but I don't know why. I'm considering the general case, the first example I gave is not on the book. I'm expecting the first way of describing lift (Wright and Cooper's) to equal the second (traditional way). The difference is, Wright and Cooper's way doesn't consider the same value for the lift coefficient. You can see by yourself by taking any lift curve and trying both. $\endgroup$
    – Sopmach
    Apr 20 '21 at 2:53
  • $\begingroup$ You will probably find a complete answer on the aviation.stackexchenge. $\endgroup$
    – kamran
    Apr 20 '21 at 4:28
  • $\begingroup$ @kamran Thank you. I'll make a post there. $\endgroup$
    – Sopmach
    Apr 20 '21 at 4:41
  • $\begingroup$ Without being certain, a possibility is that the value $ M= q\;e\ c^2 a_1\theta_0$, is indeed the moment only from the initial position of the airfoil, which can be then later used to calculate the $\theta$ (or $\delta \theta$ if you like). Then the new total (rigid+ elastic) can be used in an iterative process until convergence. $\endgroup$
    – NMech
    Apr 20 '21 at 9:23

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