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Forumula for calculating the force on a coilgun's projectile?

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Yanom

Thu Mar 07 2013, 01:35AM

Registered Member #4659 Joined: Sun Apr 29 2012, 06:14PM
Location:
Posts: 158

For electromechanical solenoids (of which coilguns are a type), Wikipedia says:

wrote ...

The force applied to the armature is proportional to the change in inductance of the coil with respect to the change in position of the armature, and the current flowing through the coil (see Faraday's law of induction). The force applied to the armature will always move the armature in a direction that increases the coil's inductance.

does this mean that for a powered coil with an iron element,
Force = (dL/dX)*I
? (note that the number of turns of the coil is already handled by the inductance L)

(dL/dX) is the derivative of the function correlating the inductance with the location of the projectile at the current location. (correct me if I'm wrong, I haven't taken Calculus yet)

does this formula (Force = (dL/dX)*I ) work, or am I missing a constant in there somewhere?

Yandersen

Thu Mar 07 2013, 03:47AM

Registered Member #6944 Joined: Fri Sept 28 2012, 04:54PM
Location: Canada
Posts: 340

IMO, it is just a qualitative approximation. Still the right way is to simulate thing in FEMM to get a precise value for force relative to coil current and system geometry.
For some reason my early simulations have shown strange correlation between magnetic energy change and kinetic energy increase, so wiki might be right, but it is not the best way to understand how the thing work - projectile pulled by force, not by knowledge where it should be to increase inductance.

DerAlbi

Thu Mar 07 2013, 07:13AM

Registered Member #2906 Joined: Sun Jun 06 2010, 02:20AM
Location: Dresden, Germany
Posts: 727

Sry to correct you, Yan, but this formula is not a qualitative approximation - its a correct analytical solution to describe the force. However i think the current should be squared! (it often mentioned that below saturation... blabla...)

F(x(t),I(t)) = dL(x(t))/dx(t) * I(t)^2

As proove: Henry/Meter*Amps^2 = Newtons.

However Yan is right that you cant use this formula since desribing the inductance in dependence of projectile position is hard (but possible) for its own.
Bringing all stuff togehter to get a correct analytical estimate if the RLC-Current waveform WITH a moving projectile develops into a higher (>3) order differential equation - which (in my opinion) will not have an analytical solution. So... you end up with iterative numerical solvers anyway... which is.... FEMM+a Lua script^^

Edit: *aaaarghhh* it really itches me to develop the analytical formulation to the problem. it is not that hard. even with external iron. This would lead to a small computer program that only solves the diff.eq. Should be fast as hell - comparing to FEMM+Lua.. sry that i have no time for that currently

This kind of stuff was basic education some semesters ago

Yanom

Fri Mar 08 2013, 03:13PM

Registered Member #4659 Joined: Sun Apr 29 2012, 06:14PM
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Posts: 158

DerAlbi wrote ...

i think the current should be squared! (it often mentioned that below saturation... blabla...)
F(x(t),I(t)) = dL(x(t))/dx(t) * I(t)^2

Oh, that's right. Force only increases with the square of current up to the saturation point, right? Then after that, it is linear with current. A more complete equation might be
F(x(t),I(t)) = dL(x(t))/dx(t) * I(t)^2 + dL(x(t))/dx(t) * J(t)

Where I is the current up to the saturation point, and J is whatever portion of the current remains past saturation.

right?

DerAlbi

Sat Mar 09 2013, 02:48PM

Registered Member #2906 Joined: Sun Jun 06 2010, 02:20AM
Location: Dresden, Germany
Posts: 727

Nature will not work like this.
The issue is respected by Âµ_r allready.
It should not be constant of course.

As mentioned before... its highly nonlinear thing.

Yanom

Mon Mar 11 2013, 12:55AM

Registered Member #4659 Joined: Sun Apr 29 2012, 06:14PM
Location:
Posts: 158

DerAlbi wrote ...

Nature will not work like this.
The issue is respected by Âµ_r allready.
It should not be constant of course.

Care to elaborate on this u_r? How could I find the force even above saturation?

BigBad

Mon Mar 11 2013, 02:27AM

Registered Member #2529 Joined: Thu Dec 10 2009, 02:43AM
Location:
Posts: 600

Yandersen wrote ...
IMO, it is just a qualitative approximation. Still the right way is to simulate thing in FEMM to get a precise value for force relative to coil current and system geometry.
For some reason my early simulations have shown strange correlation between magnetic energy change and kinetic energy increase,

Such dry humour. As if you're not sure whether conservation of energy really applies to your coilgun?

"Surely you are joking Mr. Feynman!"

wrote ...
so wiki might be right, but it is not the best way to understand how the thing work - projectile pulled by force, not by knowledge where it should be to increase inductance.

I usually find that any way of looking at something that buys you a simple way to see something is well worth pursuing. Energy arguments can be stupidly powerful, particularly when spatially non linear forces, like magnetism, gravity, are involved.

There are actually big advantages from having the forces already spatially integrated up for you on a plate.

work = force x distance

and all that.

So if (for example) you only need to calculate start and end positions and calculate energy at each, that would be super good wouldn't it?

Yandersen

Mon Mar 11 2013, 05:21AM

Registered Member #6944 Joined: Fri Sept 28 2012, 04:54PM
Location: Canada
Posts: 340

No, BigBad, I wasn't joking, I just did not put all details into the answer and you understood it wrong way. If you would ever try to make a simulator in FEMM you will understand.
The simple simulator assumes simple circuit: cap, SCR, coil. It models the discharge of the cap and change in current - step by step. Inductance of the coil is updated each step too. Each step current changes and force moving projectile too. Energy dissipated is R*I*I, kinetic energy change is Force*distance, cap's energy is C*U*U/2. Assume simulation is over when current ceases to 0 (cap is inverted). If you total the energy (dissipated, kinetic and cap) you will discover that it is higher than it was initially - exactly by the value equal to kinetic energy increase (just like this energy was taken from somewhere externally or it simply regenerated magically).
The mistake is that current was not recalculated according to inductance change: if inductance increases, current should be decreased so the total L*I*I/2 will stay constant. After this correction is done, the kinetic energy increase should be substracted and current should be recalculated again. In this case energy total will be right.
Are you following, BigBad? That is what I mean, understood? :P
So judging kinetic energy increase from inductance change analytical way IMO not possible due to dependency of inductance on current and position both at the same time.

BigBad

Mon Mar 11 2013, 09:22PM

Registered Member #2529 Joined: Thu Dec 10 2009, 02:43AM
Location:
Posts: 600

But what does the Lagrangian look like?

Yandersen

Mon Mar 11 2013, 09:54PM

Registered Member #6944 Joined: Fri Sept 28 2012, 04:54PM
Location: Canada
Posts: 340

Are you asking me?! Read my signature.

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