# How much force is produced by control surfaces?

## Context:

For some context, I'm a game developer and I'm building a flight sim game. My goal is to have realistic -- not arcade -- physics. The game is in Unity. Unity handles the actual application of the forces -- I just calculate the numbers.

The forces applied to the plane are:

• $$F_T = \text{Thrust}$$ applied to both engines directed forward relative to the plane
• $$F_L = \text{Lift}$$ applied to the center of lift directed perpendicular to velocity
• $$F_{RR} = \text{Rolling Resistance}$$ applied to the center of mass and directed opposite to velocity
• $$F_W = \text{Weight}$$ applied down at the center of mass
• $$F_{AR} = \text{Air Resistance}$$ applied to the center of mass and directed opposite to velocity

The center of mass is positioned slightly forward of the rear landing gear. The center of lift is slightly behind and above the center of mass.

I use a constant thrust directed forward relative to the plane, a constant weight force directed down (world), lift calculated with the below equations directed perpendicular to velocity, and drag directed opposite to velocity. I also have a down force provided by the elevators.

My plane is loosely based on an A320 using a wide variety of figures found online. The mass is $$72,000 kg$$, wing span is about $$35m$$, wing area is about $$122m$$, engine thrust is $$110,000N$$ each, rolling friction coefficient is $$0.04$$. The lift coefficient is taken from a table graphed against angle of attack. It looks something like the Cessna graph shown here. The drag coefficient is calculated based on the lift coefficient (formula shown here).

Here is a screenshot of the plane and the forces. Pretent it's an A320 -- it's just a placeholder for now. At the time of the screenshot the plane was travelling at $$150kn$$.

## Issue:

When I apply full throttle to the engines, the plane accelerates as normal. However, upon reaching a typical airliner rotation speed of $$150kn$$, precisely nothing happens. The plane doesn't lift off the ground until reaching almost $$300kn$$ where $$F_L$$ finally overcomes $$F_W$$. The way I see it, there are two possible causes. First is that my lift maths is wrong, and second is the force provided by the elevators is wrong. I calculated lift as shown here in my other question.

This begs the question, how much force do the control surfaces provide? In particular, the elevators. I know the horizontal stabilizer acts essentially as an upside down wing where the elevators act like flaps/spoilers. I tried using the same lift equation but the force was much to high and the plane would spin uncontrollably on the spot. I also tried calculating (estimating) by hand the torque required to overcome the torque lift of the wing due to the center of lift being behind the center of mass, and then a bit extra to make the torques unbalanced causing rotation. However this was not enough force.

Similarly, how do I calculate how much force is provided by the ailerons, flaps and spoilers -- how do I model this mathematically? Does it just modify the coefficients of lift/drag or do I apply a whole new force?

• How much fidelity do you want to get? Seems like you are treating the airplane as a point mass, so you are missing the three rotational degrees of freedom. – Jimmy Sep 12 at 1:30
• Well that's the thing. I'm only treating it as a point mass by applying forces to the COM because I don't know where else to put them... – mr-matt Sep 12 at 1:31
• It sounds like you are trying to develop something similar to FSX, which models based on force values and axis locations etc etc. Xplane does it a completely different way, by calculating the aerodynamic forces on each square inch of the entire body and deriving behaviour from that. Xplane is considered a much more true to life simulation and is used by ultralight and light sport manufacturers as a "poor man's wind tunnel". – John K Sep 12 at 3:19
• @JohnK Interesting! Where might I find the maths related to this? I'm not really sure what to search for. – mr-matt Sep 12 at 4:08
• Try Xplane coding or Xplane modeling, Xplane development, or something like that. Just plug various words behind Xplane. You will probably stumble into a developer forum or some such eventually. – John K Sep 12 at 4:13