Accelerating a ball bearing around a track using electromagnet

[d4rr3n]

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I am looking to conduct an experiment in which I use an electromagnet to accelerate a ball bearing around a track (image above is a good representative, only that I will be using one coil and my track is oval not circular) I want the track to have a minimum circumference so that the orbital velocity (number of orbits per second) will be maximum. My question is what is the minimum radius you can accelerate a ball around without excessive deceleration due to friction ie ratio of bearing radius to track turn radius? I guess its also a question of efficiency, the larger the track you use the more power you will have to supply to the electromagnet to achieve one orbit of the ball, the smaller the track the less power needed to achieve one orbit but it is limited.

[JAlberts]

Actually there is no restriction on the minimum track diameter due to friction because the radial loading to hold the ball against the outer wall is the same at all diameters; and, the smaller the track diameter the less ball velocity required to achieve that force. Also, as long as the ball diameter is smaller than the track tube's X-sectional diameter and track diameter which is inevitable in this arrangement, there will always be an essentially point contact between the ball and the tubing and a hard sphere rolling against a hard surface has essentially zero rolling resistance.

[d4rr3n]

As it is an oval track the velocity of the ball alternates and therefor the radial loading (centrifugal force) holding the ball to the outer wall of the track wont be constant. I just cant imagine a ball being able to turn a corner equal to say its diameter without massive deceleration due to friction. Another example, those guns with curved barrels designed to shoot around corners. You would assume if the curve in the barrel was too tight the bullet would simple blow the barrel in half at the bend therefore they must have had to make several barrels until they worked out which ratio (bullet length to radius curve) was correct.

[JAlberts]

Friction in the sense you are thinking of it is a result of sliding contact so unless there is some outside force affecting the ball's rotation then there will not be any friction between the ball and track surfaces. Side Note: The friction losses in ball type bearings are due to the friction between the balls and the balls' spacer ring or between contacting ball surfaces, depending on the bearings design, and not due to any friction at the ball race contact points; however, these bearings are subject to the same contact stress deformations due to high loading discussed below.

On the other hand, there are two other issues that can affect your ability to accelerate your ball to high velocities.

The first potential source, assuming the velocity of the ball an its resulting centripetal force are very high and/or the yield strength and modulus of the ball and/or track are low, is compressive deformation of the ball and/or the track surface at their contact point which will result in a loss of energy due the work of material deformation. This is why ball type bearings are made with hardened balls and races; and is something you should take into consideration in selecting the material for your track, particularly if you plan to use a plastic material for the track. The lost energy is in the form of heat which can result in surface heating of the ball and track contact lines that will subsequently reduce the yield strength and compressive modulus of those components.

The second potential impeding source will be aerodynamic drag on the moving ball due to air in your track tube (this is why high performance high speed energy storing flywheels are always enclosed in a vacuum container.

One other issue I will mention just to cover the subject is that if you plan to accelerate your ball to high velocities you will suffer considerable radial load imbalances in your assembly due to the ball's rotating mass and changes in velocity due to your planned oval shaped track so be sure your track assembly is well anchored (Air powered industrial vibrators are made using a rotating ball in just the manner of your planned assembly). One method of reducing this effect would be by reducing your ball's mass by using a hollow ball; but, in this case having the ball be of a high strength hardened material to resist contact loading deformation will be even more important.

[d4rr3n]

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[d4rr3n]

My concern is that as the track is oval the radial force is not constant and therefor there will be areas where the ball presses into the wall of the track with grater force then other areas along the track leading to wear in those areas and points of greater heating. We can think of an oval as consisting of 4 arcs (two with a large radius and two with a small radius). If we have an oval with similar X and Y lengths we could say that the difference between these arcs are small but if X a Y lengths are vastly different this translates to grater difference between arc radius. This must translate to grater differences in ball velocity which in turn results in rapid changes in ball momentum since P=mv. Lets take it to the extremes to illustrate the point I am trying to make, imagine I have a track which has a X length of 1 meter and a Y length of only 50mm, that is almost a sudden 90 degree turn at the smaller arcs. I don’t believe a ball accelerated down such a track would not experience massive deceleration on those bends and loss of momentum. Infact I find it hard to imagine the ball not just striking the wall with force at that bend and losing almost all of its momentum. This is what I meant when I originally talked about minimum turning radius.
We could also think of a cannon with a 90 degree bend, if the bend was gradual I can imagine the cannon ball turning 90 degrees and exiting the other end with high velocity. If the bend was sudden I can imagine the cannon ball impacting at the bend and the cannon being blown into two. I cannot imagine the ball exiting such a cannon with the same velocity as the cannon with a gradual bend.

[JAlberts]

With the friction issue behind us, we are now on the same page and you are correct that a shorter translation curve will cause an increased contact loading, stress and potential deformation of the contact surfaces and result in lost energy and momentum.
It will also result in higher reaction loading on your assembly structure and its mounting.
Regardless of the curvatures used in your assembly, a very real caution you need to be alert to when testing or or running your device is that with the cyclical loading your device will exhibit there is its ability to couple with the resonant frequency of elements of its own structure and/or its mounting base which will result extreme loadings on those elements far beyond the simple momentum transfers.