FYI, I did some playing around recently, albeit with a much smaller armature (small solenoid valve) operating in ~5ms. By modulating applied voltage (as PWM into a transistor, with catch diode to circulate current between pulses), I was not able to affect the bulk of the swing -- the force just goes from "barely moving" to "train crash imminent" in too little time to control. In addition, operation was too unreliable to do by open-loop control, anyway: due to friction and wobbly motion and whatnot I assume, the timing varies +/- several ms from cycle to cycle.
An active circuit may be feasible; the armature motion is sensible as the coil current dropping momentarily. A linear circuit (or at least a moderately fast PWM (>20kHz say) -- with a wide output voltage range, including negative*) can servo on this, significantly reducing coil current once the drop has been detected, and then perhaps rebounding slightly over time as it settles down into whatever holding current should be.
*Normally, coil voltage is only allowed to reverse by Vf or so (clamp diode). If you use a zener in series with the diode, this can be raised, albeit at a significant hit to efficiency -- unimportant for infrequent, momentary operation, but significant if you're using, for example, current reduction in the holding state. Maybe in that case, the zener can be bypassed once the pull-in cycle has completed.
So, a mechanical solution is definitely the better option.
Regarding a shorting ring, it might help, but keep in mind the magnetic field isn't following the armature like in an induction motor, it redistributes through the material fairly quickly; the point of course is to get some of that induction behavior, but it may need more than one ring. As a fairly extreme example, consider an armature completely plated in copper (pretty heavy copper at that, like 20+ oz): it doesn't matter from which direction the field is coming, it has to go through the copper "armor", which takes time (field strength and direction literally soak in, diffusing through the layer). This roughly limits velocity proportional to applied current, I think.
And the armature being something like solid steel, does help a bit -- the skin depth is quite shallow, so it may take some 10s of ms for field to reach the center of the armature -- but there's still plenty of force available just by magnetizing a fairly shallow surface, to get it going dangerously fast. Obviously, this is the baseline condition (more or less), so, whatever it's doing, it's not enough and something more is needed.
And keep in mind, a solid layer of copper cladding would increase the distance to the armature core, increasing reluctance, reducing the maximum holding force. The copper acts like air gap at DC*. I suppose ideally, there would be rings on the armature, evenly spaced along its length (ribbed for her pleasure?!...), so that field can only flow down it, to any given position, at the combined time constant of all those shunts.
*Technically, a little worse (diamagnetic), but not by nearly enough to care.
All to say, I don't think a single ring is enough, since whatever core is ahead and behind that ring can still be freely magnetized -- but restricting it section by section, with a stack of rings, evenly spaced, ought to do the trick.
There's also magnetic field change in the stator, as the incoming armature changes the intensity and angle of the field in it; so, similarly, rings there will help. That may be harder to arrange, though.
In combination with some mechanical abatement (rubber bumpers, dashpot?*), there should be a good combination in here.
*Most solenoids, as I recall, have a modest clearance fit for the stator around the armature (air gap), and have a hole in the end to relieve gas pressure. Perhaps tightening these tolerances, closing the restriction (so it acts somewhat as a gas spring -- pushes back harder as the armature comes in tighter -- perfectly opposing the rapid increase in force as it closes, perhaps?), but not completely, so that most of the air squeezes out just as it begins to rebound. If it can be sealed, it might also be filled with (thickened?) oil, to get more shear between armature and stator wall, and better control the nozzle restriction. (Dynamics will also be different with relatively incompressible fluids; analogous to, instead of an optimally dimensioned RLC snubber, just throw on a big fuckin' electrolytic and let its ESR dominate the discharge rate.)
Edit: That said, it is true that reluctance of the total magnetic path is falling, thus its field strength increasing, so a shorting ring can be applied anywhere -- but be careful of where this is. It has to make a shorted turn around the core. Usually, the stator is a serif-U-shaped steel frame, with the serifs pinched in around the armature, where it enters (and air gap transfers field to it). The path is closed when the armature seats fully inside the 'U', making a sideways figure-8 magnetic path, with the coil wound through the holes of that figure. Just like any E-core based transformer.
But then, the coil is already in precisely that location -- any shorts added around the core, would be exactly equivalent to (i.e., in parallel with) the coil acting as a short, which, you usually apply a fixed voltage to, so, it is.
The coil usually has a fairly modest electrical (L/R) time constant, so there is that; but any shorts you put around the core will be using less metal, how can they hope to have a longer time constant? Well, material can be thicker and denser at least, but it'll take a lot -- on the order of the skin depth at that frequency -- so, for actuation in the 10s of ms range, we're talking... really thick copper? Well, width counts for something as well, recall the demo of a magnet dropped down a not-terribly-thick copper pipe; not really sure. Well, probably the "full metal jacket" armature I imagined earlier, wouldn't work out too badly, but how much you can go down from there (in terms of ring width and spacing, as opposed to the "all width and no space" of a solid wrap), not sure.
In any case, it's definitely not going to do anything, wrapping the outside of a solenoid with copper (outside the legs of the 'U'); just in case that idea came to mind. There should be fairly low external field from a well made solenoid. This is all about internal fields, you'll need to work pretty invasively, wrapping parts with solid copper or whatever, to do anything.
Also a note on this post, I'm making no attempt to write a coherent structure; I'm just reasoning it out as I go, stream of consciousness. Expect earlier overly-simple statements to be refined by later insights. Or forgotten, as the case may be..
Tim
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