sizing servos for a cnc lathe

unless rotor speeds are extremely high, 10k rpm, I am going to go ahead and say that the reacting load in the carriage, is drastically higher, calculate the deceleration G's and acting mass from that in a direction change. Rotors inertia is a fraction of it. The carriage on my lathe weighs nearly 4,000 lbs, that tiny little rotor in that 7.5 hp motor has nothing in way of mass compared to accel/decel of 2 tons of carriage at say 4-5gs, that's we we have to use gear reduction of 5:1 with the lead screw just to attempt to control it.

You may very well be right, and you're certainly right in that if you use a 5:1 reduction then rotor inertia won't be a problem since inertia is transformed as the square of the reduction - so load inertia as seen by the motor will be reduced 25 times in that case. (Well, minus the inertia of the gearing itself.)

Just out of curiosity, what's your ballscrew lead, diameter and length? And what's your target top velocity for the axis?

I am fairly sure the lead screw is around 50 or 60mm, I haven't measure it, I did measure the pitch for setting up Linux CNC. I would be VERY happy with 200ipm as a top speed, I will have to test and see what the machine can handle reliably. I bought big drives, specifically to allow for strong accel and decel control.

Okay, so assuming your lead screw is 50 mm dia and 2 m long, its inertia should be about 100 kg×cm². (Twice that at D=60 mm!)

The inertia of the load as seen through the ballscrew depends on the ballscrew pitch. 1800 kg of load at 20 mm/turn becomes about 180 kg×cm². At 10 mm/turn it's only about 45 kg×cm². If my spreadsheet math is correct.

At 200 ipm (~85 mm/s, but let's say an even 0.1 m/s), a 1500 rev/min servo and a gear ratio of 5:1 I'm guessing your ballscrew pitch is about 20 mm/turn?

it's measured actual ratio I found was roughly 4.5 thread per inch. I was manually turning by hand just to get in the ballpark. The screw is totally enclosed and it's hard to get at it, and that all adds up as the rated speed of these motors is 1500rpm/ AFAIK the rapid speed of this machine is around 300ipm and max velocity of 150ipm factory. If you find better specifications let me know. it is a 1979 cincinatti milacron cinturn 12 u 22.5 swing x 60z in inches 

Yeah that makes sense. 4.5 threads/in is suspiciously close to 5 mm/turn which is pretty standard, so I'm going to assume for the moment that that is the correct value. That also gives very close to 300 ipm (0.125 m/s) top speed with a directly coupled 1500 rpm motor.

The reflected inertia of 1800 kg with a 5 mm pitch screw is only 11.4 kg×cm². IF your screw is 50 mm dia and 2 m long the screw inertia will swamp the inertia of the carriage at ~100 kg×cm². You mentioned 4-5 gees of acceleration earlier but that would require well over 500 Nm of torque with such a screw! 1 gee seems much more reasonable (taking just 13 ms from zero to full speed) but even that requires 134 Nm of torque. A 7.5 hp, 1500 rpm servo has ~35 Nm of torque so I guess you'll have to dial it down a bit more. (If your screw dia is 30 mm things look ok though at 1 gee.)

The inertia ratio looks fine at well below 3:1 probably (depending on the motor).

(I've updated my spreadsheet to separately calculate torque required for acceleration and machining.)