[Structural Dynamics] How to model a 3D wing as a 2D Wind Tunnel Model

[WhiteWolf98]

TL;DR Summary

I need to derive the equations of motion for a 2 DOF aircraft wing model, but can't work out how to get from a 3D wing to a 2D model.

Greetings Good People,

As the title suggests, I'm having some trouble getting to a 2D model. The process is to select an aircraft (or wing model), and model it as a 2D, 2DOF wing-tunnel model.

The aircraft I selected was a Cessna 172. This had a tapered wing, which after some calculations and assumptions, I was able to model as 2D, but that's not the whole story. The idea in the end is to model it something like the following:

While I might have the 2D wing, I don't know how to approximate the spring constants. This analysis would also be for a free vibrational response (or unforced) scenario, so there would be no M_ac.

One potential lead I found was from the following site. It mentioned that the spring constants are an approximation of the bending stiffness of the wing connected to the fuselage. So, I could find the bending stiffness of the wing, and then the spring constant of each spring would be half. This would be then varied, I'm assuming, by some ratio to create an angle of attack.

I unfortunately cannot find any kind of example or application of this, even under aeroelasticity, so any help is appreciated. Thank you.

 

[jack action]

Personally, I'm not sure what your question is. What kind of example or application are you looking for? Are the following ones enough?

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[WhiteWolf98]

Ah, my apologies, I must not have worded it clear enough. I'll try again.

So I'm not making an actual model, it's purely theoretical. I've considered a wing from an aircraft, and I want to express it in the form shown in the picture above. My only unknown is the spring constants, which I'm unsure how to approximate. I can't find any example or application of a real wing being expressed in such a form. Once I have a model, I need to use it to derive the equations of motion.

The reason I say, 'wind-tunnel model' is because that's how it's titled or defined in the textbook. This particular type of, 'wind-tunnel model' is a strut-mounted model.

Just for reference, the textbook is called, 'Introduction to structural dynamics and aeroelasticity', by Hodges, Dewey H; Pierce, G. Alvin.

 

[jack action]

When I saw your figure, I instantly saw an automotive half-car model. I did a quick search on the Internet for the terms. Is this what you are looking for:

https://www.mathworks.com/help/simu....html;jsessionid=87f57d2755e5147258e5a59d7b76​

There is also https://www.researchgate.net/publication/333612328_A_Road-Holding_Index_Based_on_Ride_Dynamics_for_High-Downforce_Racing_Cars with a more complex half-car model, including wheel mass, tire spring & damping, and even aerodynamic effects. All the equations are on p.5.

 

[Benjies]

In my undergrad a while ago, we investigated "corrections" to a 2D model to allow for the 2D analysis to translate directly to a 3D model. Obviously these corrections are imperfect, however there are multiple corrections provided in this paper and you should be able to field these to get 3D values from 2D measurements.

In theory the corrections translate from 3D to 2D as well. I notice you mentioning spring constants and elasticity, however these corrections should be far simpler and not involve the spring constants whatsoever. So, if you really are hellbent on needing those values for your work/project/whatever, this source may not be your answer. Either way, hope this helps.