# Applied Math Example - 2nd Order Initial Value Differential Equation - F1 Front Wing Bending

As I said before, I will continue this Front Wing series of my Applied Math Examples. We will be following up on Applied Math Example - Double Definate Integral - Finding Lift on Race Car Wing.

I wanted to show you a practical use for Differential Equations. Most people think that Differential Equations are only possible for the average genius. The fact is, Differential Equations aren't that hard (simple ones like this, anyway). I think this idea stems from how hard math was for them, because they never saw the application for it, and they certainly don't know of any real world uses for Differential Equations.

There are many uses for Differential Equations. In fact a few days ago I accidently made one up when I was writing Applied Math Example - Definite Integral - Distance Traveled.

This time, I am going to show you how a Second Order Initial Value Differential Equation can be used to approximate the Deflection of an F1 Front Wing.

As we saw in the last article in this series, Lift is created (or in the case of an F1 car, Downforce) due to Pressure Differences on the surface of the Front Wing. We approximated that Lift and applied it to the full surface of the wing. What we can do now is figure out how much Lift there is per unit Span of the wing. It is pretty simple, just take the total Lift and divide it by the Span. This is how you get "w" in the image above. Most of the time you see this, it's as a Mechanical Engineering student doing Beam Deflection problems and it's explained there could be a shelf with boxes on it or something. Well in this case, it's a Formula 1 Front Wing with a Aerodynamic Loading.

With this known Distributed Load, we want to find how much the wing will be bent downwards.

I want to be sure you know it's not my intention to teach you how to do Differential Equations. Rather, I want to show you how they can be applied. I assume that you know how to perform these operations, or you forgot because you thought they were boring and pointless, or you are smart and have the internet. So as usual, I'll let someone else give you the actual math lesson here.

## Example - Finding Front Wing Deflection Using a 2nd Order Initial Value Differential Equation

First, lets list some assumptions:

- Isotropic Material
- Constant Distributed Load
- No twisting, just bending
- Straight wing
- Constant wing section

Here are some dimensions you'll need

- Wing Length (1 side) (L) is 1 m
- Modulus of Elasticity (E) 3 GPa (3*10^9 Pa)
- Area Moment of Inertia (I) .0224 m^4
- Downforce on this portion of the wing (F) 104N
- Hollowed out NACA 23012 Airfoil

How much Deflection would there be if there were 1000N of Downforce, the Wing was .75m long, and the Inertia was .0001m^3, and the Modulus was 2 GPa, halfway down the wing?

So you can see that is pretty useful. Obviously this doesn't direcly find the Deflection of an F1 Wing. Carbon Fiber is Anistropic, the surface is very complex, there could be a core, the Aerodynamic Loading isn't so simple, and I could go on but the point is they would use FEA on a very expensive computer. I wasn't implying this is a perfect answer, but insted a quick and easy approximation we managed to make with a simple 2nd Order Initial Value Linear Differential Equation.

I hope this makes it easier for you in the future. My biggest struggle through Differential Equations in College was not seeing the purpose or application of them. To me it was like comma splices, which only matter if you intend to become a professor teaching about them. Differential Equations have many other real world uses, and this was one of them.

If you would like a more detailed derivation, please let me know.

As always, I do this to help. This article took me several hours to write. I'd love to get your feedback (through comments, sharing, or liking) because I need to know if this is helping, or if I'm better off putting my time into better things.