How to calculate missile's radius of turn?

Question

I want to make a simple simulator game and in this game there is a missile that must hit something. The problem is, I can't find the equation to calculate its radius of turn. The game is a simulator and I searched many websites, forums, ... but didn't find it.

For example here i found a equation for airplanes that depends on bank angle and speed, or I found this equation :

$$\frac{v^{2}}{A}$$

which $A$ stands for acceleration, but I don't know how much acceleration the rocket gets when it turns.

The missile turns by changing the forward canard's angle-of-attack

Actually the problem is, I need an equation that depend on the stability (in cal) which depends on its CP & CG location and also mass, speed, missile's canard area, location of canards and missile's canard angle (angle of attack) but it seems that it is nowhere to be found.

Are there any step by step instructions to calculate it?

UPDATE

For more clarification and avoiding any confusion, here are the dimensions of the missile i want to use in the simulator game:

The missile uses rear fins (fixed-stablizer).

4 canards, 2 of them for pitch and 2 for yaw (remember, for turning, i want to use two canards for pitching so in must be considered in calculations).

Location of moving canards = 1.605 m (distance from tip of the nose cone).

Nose cone geometry = Tangent Ogive.

Mass = 136 kg

Speed = 350 m/s

** Max. Altitude =** 6000 meters above sea level

At static situation:

CP = 2.69 (distance from tip of the nose cone)

CG = 2.29 (distance from tip of the nose cone)

Stability = 1.32 cal

At speed = 350 m/s:

CP = 2.79 (distance from tip of the nose cone)

CG = 2.29 (distance from tip of the nose cone)

Stability margin calibers= 1.65 cal

Canards movement limit = 10 degrees

** Temperature, air pressure & density = standard conditions with no wind**

Any instructions to calculate it (at maximum canard rotation (10 deg., speed = 350 m/s altitude = 6000 meters above sea level and other conditions)?

I am looking for step-by-step instructions that can solve every example like this.

Notice = I want to navigate the missile after burnout, so at start, the missile goes ballistic, then, after burning out the fuel, navigation starts. So, there is not any mass changing during navigation flight.

Answer 1

I'm far from an expert on this. In fact I am not formally educated in this at all.

But I would think you need need Cl vs AOA lift curves for all your control surfaces. From there you need to mix it with your airspeed and control surface to get lift, then mix that with your missile geometry to get moments. Then you need to hunt for the equilibrium between all these to find the balance point.

You can probably simplify things by assuming constant Cd within your operating range and then pick a Cl vs AOA data for an airfoil that has a good linear section and just choose to only ever operate within that region. Then you can describe the segment of that graph with an equation. That way, you could at least your can stay working with solving simultaneous equations which I think you're going to need to solve iteratively anyways.

These airfoil plots aren't for real finite span airfoils though so if you want it accurate you may need to add in fudge factors to account for that. Particularly how the low aspect ratio of missile aerodynamic surfaces decreases the lift at any given geometric AOA while allowing it to operate at a much higher AOA without stalling.

Otherwise you will need to learn simulator equations which you will eat up all your time and you probably aren't interested in. Or entering pre-tabulated data from pre-existing airfoils. With pre-tabulated Cl vs AOA data you will end up exploding it into more mult-dimensional look up tables for multiple control surfaces once you convert them to lift, airspeeds, bank angles, and moments. Then you need to find the intersection between all of them to find the stable point, if there is one.

It's quite similar to what you need to go through to calculate the airspeed of an aircraft based on the propeller data...but more complicated because there more dimensions and control surfaces are involved.

As a test case, try doing it for just the pitch of an airplane flying level. So just two control surfaces. See if the result you end up with for the resulting AOA of the plane seems reasonable. Then you can start adding things like radius of turns which would involve bank angles and the components of the aerodynamic force of multiple control surfaces parallel to the radius of the turn adding up.

Answer 2

Hoping you're not planning to make a real missile navigator, here's a simplistic take:

1. calculate the local lift of control surfaces (canards)

1.1) take it as a bit worse than a NACA0012 and use the simplistic linear equation, CL = a0 * alpha_local. (where alpha_local = alpha_missile + control_deflection)

1.2. take a0 = 5.5 CL/radians and assume this linearity holds till a stall angle of 10 degrees. (so your controller only runs -10 to +10 degrees).

2. find the max lift =

Lift = 5.5 * 10 * pi/180

3. calculate the fin stability derivatives (or assume something) and use a moment equation,

M = M_Control_Surface + M_Fin

4. Find the trim location where

M_Fin = M_Control_Surface

This will give the alpha_missile for the whole missile.

5. the whole missile will be experiencing a lift (body + fin + canards), calculate them again with some CL_alpha relations, and add them up.

6. calculate Lift from CL equation:

L = CL_total*q

where, q = dynamic pressure.

The Lift is what is causing the whole maneuver, so use it as your centripetal force to calculate radius of turn, acceleration, etc.

This answer is probably sufficient to design the game mechanics for a simulation game. It's not scientifically correct, but games do not require real physics most of the time, and basic approximations are the way to test the playability and demo it before investing more time in improving the physics engine.

Btw, you'd also need Runge-Kutta or other integration methods to predict next step iteratively.