Formula (or heuristic) for calculating force required for incremental change in angle of attack of an airfoil

Question

Assume I have a system comprising of an airfoil A immersed in a fluid.

Further assume, that the airfoil is affixed via two bolts at one end, and is untethered at the other end.

The system can be parametised as follows:

• S is the span of the airfoil
• C is the chord of the airfoil
• V (vega) is the relative velocity of the fluid passing over the airfoil
• $\rho$ is the density of the fluid in which the airfoil is is immersed
• $\theta$ is the current angle of attack of the airfoil (in radians)
• L the distance between two bolts affixing the airfoil at one end (< C)

[[Question]]

Given a new angle $\theta_i$, what would be the formula for calculating the force required to be exerted on the two bolts to change the current angle of attack of the airfoil from $\theta$ to $\theta_i$? (assuming all other variables held constant)

[[Notes]]

From the lift characteristics of an airfoil, I think its fair to assume that a greater force will be required to be exerted as the angle of attack increases (up until the stall angle)

Additionally, since the force required is likely to be monotonically increasing (up until stall angle), I would prefer if the function actually returned the supremum of the the forces required to increase the angle of attack from:

$\theta$ $\rightarrow$ $\theta$ + $\delta$$\omega$

where $\delta$$\omega$ is the change in radians divided into an infinitesimal number of steps.

Ideally, the formula should derived from first principles (or the answer be provided as pseudocode for an algorithm), so that I can follow the logic, and apply it to an airfoil of non-rectangular shape.

Answer 1

As per "NASA Aeronautics And Space Administration" the thin foil lift equation is $$L = Cl * A * .5 * r * V^2 $$

Cl is the lift coefficient and in small angle range is directly related to angle of attack, multiplied by other factors. They have a java app here FoilSim app Which is similar to what you seem to be asking. You have to set the security of your computer Java to let this app run. They offer help to set it up.

As far as your model of the wing and its attachment to support via two bolts, it is not practical and minimum number of bolts required would be three non-collinear bolts to turn yor hinge connection into a cantilever connection.

Answer 2

Without knowing the exact geometry of the system, such as the shape of the foil, it would be hard to come up with an exact model. You might be better of doing an experiment, applying some force and record what kind of angle you get in response for different relative fluid speeds.

This force for a given angle will only be the force required in steady state, but during the transition from one angle to another you would also need some extra force to initially speed up and slow down the rotational speed of the airfoil. So you would need some time varying force, which can be achieved using some feedback control. For example a PID controller would be a possibility. The integral term of such PID controller should also help bring the airfoil angle to the desired angle in steady state.

You could also include the model from your experiment in the form of feed forward control, which could be used to improve the performance of the system.

Answer 3

For a given angle of attack and velocity you can compute a pitching moment about the cg (Or the normal force on the airfoil and the center of pressure), this moment can then be resolved onto the bolts to determine the loading on the bolts.

You'll need a controller (like a PID) to exert the loads on the bolts correctly to stop at the correct new angle of attack.

Edit: We need to first define the center of pressure where by definition the pitching moment is zero. Some math will show us that the center of pressure can be found from:

$h_{cp}$ = $h_{ac}$ - $Cm_{ac}/C_l$

where $C_l$ can be found from the lift curve slope for your airfoil in Theory of Wing Sections and $Cm_{ac}$ (Moment coefficient about the aerodynamic center) which is constant can also be obtained from Theory of Wing Sections.

Now for your back of the envelope calculation you can just assume that the aerodynamic center is roughly quarter chord from the leading edge ($h_{ac} = 0.25$).

Now we know where the Lift force is acting ($h_{cp}$) as well as its value ($l=0.5*C_l*\rho*V^2$).

Now we just need to resolve that onto the bolts using simple statics. I'm going to use a pin-roller beam for the example. Let's call the distance to bolt 1 from the leading edge $x_1$ and the distance to the second bolt $x_2$.

Summing the moments about the 1st bolt and setting equal to zero we get

$F_1 = l * x_{cp} / x_2$

and then summing the forces in the Lift direction and setting equal to zero we get

$F_2 = l - F_1$

where $x_{cp}$ and $x_2$ are the dimensional distances to the center of pressure and second bolt, respectively.

Now bear in mind this is a VERY rough estimate. This doesn't take into account 3D effects (A whole topic on its own), drag contribution to the forces on the bolts, and the fact that the beam is actually a pin-pin which is indeterminate.

For references I used http://www.dept.aoe.vt.edu/~lutze/AOE3104/airfoilwings.pdf and http://bendingmomentdiagram.com/tutorials/calculating-reactions-at-supports/ as well as some experience ;)