Force between opposing magnets

[Guineafowl]

TL;DR Summary

I’m trying to find a way to model the magnetic field between repelling magnets, and how that changes when the permeability of the gap changes.

Following a conversation with @Tom.G , I’ve come up with the following question, to which we’d both like an answer:

Imagine two drilled magnets threaded onto a frictionless upright shaft, set N-N. One fixed down, the other floating above.
Now, if I move an iron gate in and out of the gap (the shaft has a gap), the upper magnet will move down and up.
How would you express the force exerted on the iron gate by the magnets, on the in and out strokes?

For a start, I can’t find an expression for the force between repelling magnets. Is there one?

 

[Baluncore]

The expression will be related to the force between similar monopoles, unless you can give some dimensions to the magnets.

Assuming axial symmetry, you have an axial height to diameter ratio for the individual hollow cylindrical bar magnets, and a separation between the closest pole faces. You also have an inner and outer radius for the drilled magnets.

Speed and eddy currents play a part. Is a thick gate moving fast, or is it thin and moving slow?

Is the iron gate, a plate of semi-infinite extent, with a straight edge that destroys the symmetry of the model, and so precludes a sensible analytic solution?

 

[Guineafowl]

The expression will be related to the force between similar monopoles, unless you can give some dimensions to the magnets.

Assuming axial symmetry, you have an axial height to diameter ratio for the individual hollow cylindrical bar magnets, and a separation between the closest pole faces. You also have an inner and outer radius for the drilled magnets.

Speed and eddy currents play a part. Is a thick gate moving fast, or is it thin and moving slow?

Is the iron gate, a plate of semi-infinite extent, with a straight edge that destroys the symmetry of the model, and so precludes a sensible analytic solution?

I was looking at the more general question of how the work done lifting the upper magnet a distance r is transferred the the work done moving the gate.

However, if it helps, I have some drilled neodymium magnets for keeping cupboard doors shut. OD 15mm, ID 4mm (countersunk one end), length 5mm. I can reproduce the effect by threading them on a plastic shaft and inserting a spanner in and out of the gap.

Again, on the generalising front, I was hoping to ignore eddy current losses for now, to help focus on the force transfer. I have come across them in transformers and eddy current brakes, but let’s assume a very slow reciprocation of the gate.

I hadn’t considered the thickness of the gate. Max would be the size of the resting gap of the magnets, for obvious reasons. Would the minimum thickness be related to some mechanical limit, eg stiffness, or saturation? Let’s say 2mm for starters.

One way of simplifying might be to consider the static situations of gate in/gate out, before trying to tackle the transitions. Would this help?

 

[Baluncore]

Would the minimum thickness be related to some mechanical limit, eg stiffness, or saturation? Let’s say 2mm for starters.

Thickness minimises saturation, but maximises skin effect delays.
A slow-moving thin-sheet will minimise eddy currents, but maximise saturation problems.
The model may be symmetrical about the mid-plane of the plate. Does the plate remain half-way between the two opposed faces (symmetrical), or at the same height relative to the fixed magnet?

 

[Guineafowl]

Thickness minimises saturation, but maximises skin effect delays.
A slow-moving thin-sheet will minimise eddy currents, but maximise saturation problems.
The model may be symmetrical about the mid-plane of the plate. Does the plate remain half-way between the two opposed faces (symmetrical), or at the same height relative to the fixed magnet?

I’ll say the gate slides over the fixed magnet. It’s moving slow enough to discount eddy currents, and is thick enough not to saturate.

 

[Guineafowl]

I’ve found this manufacturer who has a force calculator on their website:
https://www.kjmagnetics.com/calculator.repel.asp

I’ve asked them what formula is used.

 

[Guineafowl]

I’ve had a reply from K&J Magnetics. They say the calculations used are proprietary and a result of thousands of man-hours of experimentation. There apparently isn’t a formula. That’s quite a surprise.

Would it be possible, as @Baluncore suggested in a previous discussion, to analyse this qualitatively in terms of the magnetic circuit?

 

[Baluncore]

I think you will have to build a prototype and test it.