Very true dk0, as the diameter goes down, it then gets a whole lot tougher. Since I don’t know your experience level, some of this you may already know. I’m also just trying to add some constructive input.
Depending on the actual part length versus it’s diameter, even a constant force spring loaded center while it would help still doesn’t factor in the cutting tools forces on the shaft being turned along it’s length. Standard molded carbide tips or poorly sharpened HSS are just about useless because of the cutting tool pressures required to even get the material to cut. The small shaft diameter then doesn’t have enough rigidity to resist those forces and using conventional longitudinal cutting. It would just deflect away from the tools cutting tip instead. So lapped carbide or imo better, honed to razor sharpness HSS starts becoming mandatory. There are still definite limits for that diameter verses length of cut no matter how sharp the tool might be even when your using tail stock support.
On a much larger scale and even with the best live center support, turning tapered rifle barrels are a large problem for gunsmiths due to the exact same part deflection, and worse, surface finish chatter created due to poor support over the increased distance between the head and tail stock with those long rifle barrels versus there cross sectional area and diameter. Heavy calibrated spring loaded or even hydraulic traveling steadies have been used for that to allow for the part tapers. Fixed steadies and just step cutting the taper is also done. Cnc and industrial levels of the same type of parts with it’s expensive specialized tooling and methods are well outside how we might be forced to do the same.
Since you’ve clarified what your wanting to do, I understand it a bit better now. Depending on that actual part length / diameter? Industry has solved that in various ways. Traveling steadies, or in case you don’t know of them, even what are called box tools that are generally but not always used with high volume screw machines, turret lathes, or for specialized needs on cnc lathes where tail stock support might not be used at all. The full cut depth is taken all at once. How those adjustable box tools work could still be built to work exactly the same on just about any manual lathe. While this video is about using them on a turret equipped screw machine, it does show how they work and mentions the important set up considerations. https://www.youtube.com/watch?v=V-ClQSknzWY For the diameters you mentioned, then obviously quite small bearings would be necessary if you were making one to fit your own lathe and tool holder size.
There’s another method shown quite well in this video. https://www.youtube.com/watch?v=X-TkuQDWdbA There’s also a clever shop made traveling steady in this video at about the 9:30 minute mark. https://www.youtube.com/watch?v=CDi5jTxmmG0 He’s using it to do small diameter threading, but the exact same idea could be easily used for just plain turning and small diameters. Over the years, there’s been other similar designs published in the Model Engineer and other magazines for turning very small diameters. Most seem to use shop made bushings for the support and each desired diameter. From my reading and any of the videos I’ve watched, it seems that once diameters get small enough, live or any centers aren’t used at all.
Your idea is I think certainly valid, but from my perspective and personal testing with my own lathes. It’s still assuming a mini or any lathe has the tail stock in almost perfect alignment to the lathe bed ways and head stock. And as the diameters reduce, that alignment becomes even more critical with any tail stock support for the finished part parallelism without any unwanted taper. Samuel wasn’t I think being contrarian at all, he’s brought up some very valid points. A long part of say 30 mm diameter will show multiple times less part taper than one that’s say 2 mm in diameter as it’s being cut if the tail stocks center line is either high or low. And even the most expensive and best industrial quality lathes made today are all built with the tail stock center line purposely high. Page 49 & 50 in this PDF. https://pearl-hifi.com/06_Lit_Archive/14_Books_Tech_Papers/Schlesinger_Georg/Testing_Machine_Tools.pdf That’s done for a few important reasons, to help compensate for future wear on the tail stock base, and even for large work piece weight and cutting tool pressures.
An off shore produced mini lathe may have the tail stock center line pointing almost anywhere and quite often they are. In fact my own Sieg C6 sized lathe when I bought it many years ago had the tail stock quill pointing uphill by around .009″ or .23 mm over just 2″ or 50 mm of quill extension. The upper casting of the tail stock had to be re-machined and then shimmed back to the correct elevation before it would quit breaking the tips off any center drill I tried to use. While I haven’t personally tried your idea, a traveling steady, tooling or methods used in those videos would be my preference. Those box tools I haven’t tried yet, but one of those should work even better. If the part length got very long and possible part whip became a concern, a slightly oversized hole in a bushing and made from almost any material held in your tail stocks drill chuck as mentioned in the second video could easily eliminate that from happening.
Your idea of using Belleville disk springs would make for a more compact package, Michael’s very clever find of the internal workings of those Concentric / Riten centers show a coil spring. But there’s generally a good reason why design choices like that are made. And since those Belleville springs have been around since 1867, they were certainly an available choice when the Concentric Tool Corporation invented that spring loaded center. I have to think there might have been a good reason why Belleville springs weren’t used. Cost between the two designs of spring doesn’t seem to be a real factor, so there may have been some definite design reasons for choosing a coil spring instead. I just can’t think of any logical reason for what that might be yet other than just one. A coil spring would allow more length of compression than a Belleville spring. That could or may have been the only reason.