Re: Railgun mathemathical model.
Dr. Shark, Mon Jul 16 2007, 12:00PM

Welcome to the forum, but please familiarize yourself with the rules before you post:

Part II: Posting

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EDIT:
I should have read them myself:
F. Let the moderators do their job. If you're not a moderator, you shouldn't be accusing new members of not having read the rules, or of not Googling properly. You certainly shouldn't be making people feel unwelcome by acting hostile. If there's a problem with a post, report it to a moderator, and let them handle it.


so disregard what I said. I don't know about a FEMM model (I'd sure be interested!) but in the meantime have a look, there is some stuff here:

Re: Railgun mathemathical model.
WaveRider, Mon Jul 16 2007, 07:44PM

You do not need FEMM to simulate a railgun... In fact, to first order, ou can use the concept of a shorted "twin-lead" transmission line to simulate it. Google a bit, consult a good text on EM theory, and if you still need help, send me a message....

Re: Railgun mathemathical model.
Eric, Mon Jul 16 2007, 08:38PM

It is very simple to create a 2D cross sectional model of rails in FEMM. That gets you an accurate number for the AC/DC inductance gradient (L') of the rails and therefore the projectile force: F=1/2*L'*I^2.

Naturally a 3D simulation would be much better but the 2D simulation will still tell you some important things about your geometry.

Here's a very simple geometry in FEMM.

Re: Railgun mathemathical model.
JoErnst, Wed Jul 18 2007, 09:29AM

Ok,thank you,WaveRider, but I'm yet studyng at colledge =) and can't make EM calcs.

Re: Railgun mathemathical model.
WaveRider, Thu Jul 19 2007, 05:44PM

JoErnst: You don't have to do any EM calcs (altho' if you are asking about FEMM, that is precisely what you would be doing... :)

All that you need is an estimate of inductance-per-unit-length of rails (which form a transmission line). FEMM can give this to you or you can estimate it by, for example, assuming that the rails are of circular cross-section (there is a closed-form mathematical expression for this case). The force will simply be


Note that the inductance-per-unit-length is usually a fairly small number. That is why you need lots of amps to get significant accelerating force!
Cheers!

Re: Railgun mathemathical model.
badastronaut, Mon Jul 23 2007, 07:45AM

If it is not important for very precise numbers, then you can assume that the inductance gradient for any simple railgun is .5uH/m. This will give you a ball park estimate of force.

Keep in mind that even if you had zero friction with a very low mass projectile in conjunction with an enormous peak force, say 10 MN, you still might end up with very low projectile speed.... Final speed at the muzzle is a function of total net impulse delivered to the projectile, not force.

FEMM is a simulation tool. It will not give you a mathematical model, but you can derive relationships based on your simulations. If you know enough to use FEMM, then you will have the skill to draw some blocks, add labels, and run the simulation for a static field. A better simulation will use Lua scripting.

If you understand how a railgun works, you can make your own model. I would suggest a coupled system of non-linear second order differential equations. Take into account Faraday's law because that will allow your model to follow conservation of energy.

May I ask what you need the model for?

--edit
The thread linked by joe has the railgun efficiency equation. I believe it is accurate and a good model. It is actually the closed form solution to efficiency for the non-linear differential equations I was talking about, but you don't need to know anything about DE's to derive it. ;)

That is about as simple of a model you can get for a railgun: a single plug and chug equation. Though, you won't be able to learn anything from the equation by itself...

Re: Railgun mathemathical model.
WaveRider, Mon Jul 23 2007, 08:31PM


FEMM will give you the inductance/length, as I indicated before, which you can use in the DEs that badastronaut has mentioned. The field problem is indeed linear (since there are no saturable materials present, assuming perfect connections between rails and armature, etc.)

The static field solution that FEMM would give you is probably valid since the mechanical movement of the armatuire (changes to the electrical circuit) are of time scales much larger than the EM wave-propagation characteristics along the rails.

You should have equations like this (simplified, in the absence of friction):

1. dv/dt = F/m
2. F=0.5 * L' * I^2 
3. V = R_rails(z) * I + v * L' * I + L' * z * dI/dt  
4. -C * dV/dt = I

• 
  L' = inductance/length
  m = armature mass
  v = velocity
  F = force
  R_rails = resistance of rails/armature/firing circuit
  V = Voltage across firing capacitor
  C = capacitance of firing capacitor
  z = position along rail
  

The first equation is Newtons law. The second is the force relationship to current. The third is Kirchhof's voltage law (which contains Faraday's law). The final one is the rule that relates the current through to the voltage across a capacitor. If you want to get get really sophisticated, you can have R_rail vary in a stochastic way to simulate imperfect contact between the rails and the armature as it moves... ;)

This should be qualified. Impulse delta P depends not on a specific instantaneous force, but on its average over the firing time, i.e.


In terms of rail length, it is often more convenient to write this as

where A is the constant of integration...