For a drive axle length of ten feet, I’d probably do it in sections and ise pillow blocks to stabilize it and some type of flexible coupling to join the sections. U joints, lovejoys, thomas coupling, etc.
I’d also probably forgo trying to turn something that long on a home lathe and instead go with a structural tubing and welded ends. Structural tubing isn’t cheap though. I bought a 5” length for a motorcycle frame head/neck and it was nearly 100 bucks. In all fairness, the diameter was a lot larger than 1”.
You can probably get by with a more standard wall thickness if its just transferring power, like a vehicle driveshaft does. If its the actual screw mechanism that moves feed, you need to re-evaluate if tube is strong enough for the task. The problem with tube at 1” is thinner walls won’t have a lot of strength.
A feeder is probably a low speed shaft so you likely can get by without balancing them.
Even if its in sections, I believe you’re going to have trouble turning to consistent tolerance (Ie: tapering) across the shaft on a home sized lathe. Longer lengths can “deflect” in the center of the span unless a follower rest is used. Most home lathes are limitied to around 24”-30” between centers and not many are going to have a 1” spindle hole to accept longer lengths. Keep in mind that a 36” lathe won’t fit a 36” piece between centers. The legth is the total length of the lathe bed and you loose working length for the headstock and tail stock. so a 36” lathe might inly do 24” between centers. My Atlas TH42 tops out somewhere around 30-36” (i’ve never used it to max length).
If you want to use solid turned shafts, I’d suggest “farming out” the work to a fully featured shop and have them do it. They should have proper gear to turn long lengths to tolerance and Have machines big enough to pass 1” stock through the spindle hole.
Just my .02….
