I am currently working on a flight simulator and I try to simulate every actions that applies on my plane. As you know, planes tend to orient themselves into the airflow direction, because of the "weathervaning effect". It's the same effect that explains why arrows stay straight in the air. So let's take an arrow for example, with an angle of attack of alpha degrees.

The wind force that applies on the fins of the arrow is :

$$F=\sin\alpha \cdot\ K$$

With $K$ a coefficient changing with air density, airspeed, drag coefficient and so son.

So the torque generated on the arrow is :

$$\tau=\sin\alpha \cdot K \cdot d$$

With $d$ being the distance from the back of the arrow to the center of gravity, on which the arrow will rotate.

When I run this into my simulation, the arrow realign itself onto the wind, but... It overshoots everytime. When it overshooted, then it tries to realign going the opposite direction, but overshoots again, and so on.

And, of course, that's obvious. Because there are no energy loss on that simulation.

My question is :

Why do in real life, arrows will never oscilate so much? What force, linear or angular drag causes it to not overshoot and align into the wind? How can I simulate that to obtain a valid simulation?

Thank you.

  • $\begingroup$ If you search the web for "physics of arrows" you'll get a lot of information. I get the impression you're talking about what is known as "The Archer's Paradox". $\endgroup$ – StephenG Apr 23 at 13:14
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    $\begingroup$ Thanks, but I'm not talking about the Archer's Paradox at all. I am considering the arrow straight and that cannot bend. I am talking about the aerodynamics of the arrow. My example works as well with a weathercock but I cannot find deep explainations rather than "wind strikes the tail of the weathercock, and it aligns on wind." Ok, but using this model the weathercock will accelerate toward wind direction, and would overshoot as there are no energy loss. So something is missing $\endgroup$ – Anselme Apr 23 at 13:20

Does your simulation include friction? Friction contributes a damping term to the equations of motion.

How about inertial effects? In order to oscillate, an arrow needs an excitation. With the high moment of inertia along axes perpendicular to the flow direction, it requires a substantial excitation in order to produce a noticeable oscillation. Where should that come from?

What you describe sounds like an idealised case without friction and some arbitrary starting condition that is unlikely to be realistic.

But in aircraft this really happens. An excitation can be the pilot stepping on one rudder pedal and then taking his feet off them, so you experience lateral stability with loose rudder (hint: letting the rudder float freely reduces stability). The old certification requirements for GA aircraft (FAR 23.181) demand that the ensuing oscillation must die down to a tenth of its initial amplitude after seven cycles.

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  • $\begingroup$ Hi, that's what my question is all about. The case I described is simple and idealised on purpose. Because I don't know what forces and how they act on my arrow are to be taken into account, otherwise that the upcoping wind. Can you explain more deeply how friction works to dampen the yaw instability of my arrow ? Because friction is already what is making my arrow oscillate. $\endgroup$ – Anselme May 3 at 12:48
  • $\begingroup$ @Anselme: Drag on the vertical is pulling it always towards the center. Same on the wings: The forward moving wingtip experiences more drag. So any motion causes drag that impedes this motion. The forces involved are small, so they need time to become effective. $\endgroup$ – Peter Kämpf May 3 at 15:21
  • $\begingroup$ Ok, I think I start to see. Thank for the help $\endgroup$ – Anselme May 18 at 19:22

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