On a small lathe, you should use the chuck and steady rest to support the ballscrew. This sort of double ended lathe support helps to ensure that your center hole will be centered with respect to the axis of the screw. Holding by the chuck only is no assurance that the back end of the part is properly aligned with the lathe axis, and the center hole may end up to be off center. Off center "center holes" are a real pain in the a$$ to try to correct!
I am not sure what kind of surface the ballscrew has to run on in the steady rest. You DON'T want the ballscrew threads to catch on the steady rest and start pulling this way or that. You might be able to find an off the shelf sintered bronze bushing that will slide over and fit your ballscrew tightly enough to serve as the bearing surface to run inside the steady rest jaws. Or, shim the bushing so that it is tight on the screw, and still evenly centered over the screw.
Heat the end of the screw to try to anneal it for the drilling. If you are lucky, a very faint red heat will be sufficient to reduce the hardness sufficiently. Just let it air cool. Just heat as much length as you will need to rework.
Drilling a center hole in the steady rest seems difficult at first. First, center the part by adjusting the steady rest jaws so it looks halfass close to centered with respect to your tailstock center. BTW, your steady rest should be very close to the end of the shaft as you do this, to minimize deflection.
Some beginners like to "dial in" their part in the steady rest which is a complete waste of time
Of course, any round part will turn in its own center, using the exterior as the bearing.
The trick is to carefully approach the nicely squared and faced end of the part with the center drill. As the part is rotating under power, bring the tip of the center drill into light contact with the end face of the part. If it scribes a little circle, then stop and readjust the steady rest jaws to bring the center drill point into the center of that circle. Now drill in just enough that the pilot end of the center drill has only begun to enter at full diameter. It may even be roughing out a slightly oversize circle at this stage, but do not drill in past the angle of the drill pilot point yet.
Stop the spindle again, but leave the center drill engaged in the hole it has started to make. At this time, you can carefully go around and slack off all the steady rest jaws. You may find that one of them seems to have a lot of pressure on it. This is because the setting was a little bit off. Now, adjust all the steady rest jaws back up till they touch lightly. Then carefully turn the spindle on slow, leaving the center drill still acting as a crude "tail-center" in the end of the part. Go around and gently tweak the steady rest jaws more firmly onto the part so that they all have equal pressure.
Now you are ready to finish drilling the center hole, so do it.
Keep your long hair, beard, necktie and shirt sleeves away from the rotating machinery, please. ![Smile](images/smilies/smile.png)
Edit: Swede posted while I was preparing this. No offence intended, Swede. This is just a different way. ![Smile](images/smilies/smile.png)
I like Swede's suggestion of custom making the bushing to go over the screw.
First you get good, then you get fast. Then grouchiness sets in.
(Note: The opinions expressed in this post are my own and are not necessarily those of CNCzone and its management)