Hi All,
here we go down the rabbit hole again! A couple of observations:
First, never enter this rabbit hole unless the lathe is not producing the parts you need to the accuracy you require.
With an engineers level, check across the bed at either end, and shim the cabinet untill the readings are as near identical as possible. you need to shim at four points, front and rear, at the headstock and tailstock end. The idea of this is to twist the bed into as near perfect alignment as possible. this process is best done over a few days to allow the bed to settle, and then be checked again, when you are happy with this, proceed as follows
It is pointless to try to align using the three jaw chuck, the very best of which are usually at least 3 thou out WHEN NEW!
Remove the chuck, check and clean the spindle bore, fit the bush and a known good centre, and then put a DTI on the spindle, and then on the centre.
If there is run out on the spindle, tighten headstock bearings to eliminate it.
If there is runout on the centre, remove the centre and bush, check the bore again, check the bore in the bush, and re seat the bush and the centre.
Buy a test bar,, remove the centre from the headstock and fit the test bar in its place, and check with dti at the remote end, you may need to remove and reseat the test bar till you get the absolute minimum possible run out.
Place the DTI on the saddle with its foot on the front of the test bar at centre height, and wind the saddle from the headstock towards the tailstock, If the headstock is misaligned you should see the DTI needle move steadily in one direction, if the reading is erratic, you may be looking at bed wear.
Take note of the runout at the remote end, rotate the spindle 90deg, and try again, note the reading and repeat till you get the spindle back to its original position.
You should now have a good idea about headstock to bed alignment, and will be able to make a decision as to whether you need to realign the headstock front to back, and whether the bed is worn or not
Now move the Dti to the top of the test bar and test from headstock towards the tailstock to see if the test bar is parralell to the bed, again be aware that you are looking for a constant downward or upward trend, and any erratic movement in the needle may be bed wear.
You now have a good idea whether you need to shim the headstock to correct any "nod" in the alignment.
You will usually find that by slackening the headstock bolts slightly, and bumping the headstock with a soft faced mallet in the desired direction, front to back alignment can be corrected.
Sometimes, selective retightening of the bolts can correct up or down nod, but if not you will have to shim, or scrape, but it may be a case of removing the headstock, cleaning the plate it sits on, identifiying and removing any dings or burrs with a flat stone or small scraper.
Now you have the spindle sorted and the headstock alignment as accurate as possible, you can move on to the tailstock. Whilst any of the above tailstock methods will work, I usually put my best centre in the tailstock, and with a dti on the side of the test bar, wind the centre into the centre drilling whilst watching the reading. Any discrepancy here can be corrected with the tailstock alignment screws.
Now repeat with the DTI on top of the test bar, and you will see if you need to shim the tailstock, by how much, and whether it needs to nod down or up.
Now move on to the turning tests ans see what results you get!
I must add that this is the way I do it, based on as much logic as I can muster at the particular moment. I am not suggesting that you go this far, indeed, you should only correct a fault if it occurs, and not go looking for trouble, because it is all to easy to imagine there is a problem. As I said at the top, only dive down the rabbit hole if your machine is unable to make the parts you want to the accuracy you require, and I will add to that that most lathes and associated chucks/tooling etc benefit from a complete strip down, clean, debur, lubricate, and carefull reassembly
The best of luck!
Phil