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There generally isn't that much difference between an axis servo motor, and a spindle servo motor. The main difference is in how they're utilised.

A servo motor rated speed/torque is what it should be able to happily produce 100% off the time without any overheating problems.
In terms of an axis, you're never likely to sustain using 100% of that continually, so you can intermittently drive the servo harder, which is where the instantaneous torque figures come in.

The rated speed is where the laws of physics really kicks in.
Regardless of being AC or DC, motor torque is directly proportional to current. More current = more torque.
In the case of DC, motor speed is proportional to voltage. More voltage = more speed.
In the case of AC, motor speed is proportional to frequency. Higher frequency = more speed.

As a motors speed increases, the back emf (electromotive force) increases. Think of emf as if the motor was acting like a dynamo/alternator being spun, in that it produces power, aka emf. The faster the motor spins, the more emf it produces. Now this emf fights 'back' against the voltage being applied to the motor, so the faster you spin the motor, the more voltage you need to maintain a given current through the motor. This applies to both DC and AC motors.
At a motors rated speed, this is the point the rated supply voltage can still overcome the back emf to provide the rated current.

Above this point, with an AC servo, as speed is not reliant on voltage, you can continue to increase the frequency, but you then start to lose torque, as you don't have the voltage to force enough current through the motor.
The motor essentially goes from being a constant torque source, with power proportionally increasing as motor speed increases to the rated speed, to a constant power source, with torque proportionally dropping as speed increases.

This applies to pretty much all electric motors.

In terms of a standard servo, and a spindle servo. The main difference is likely to be the spindle servo is derated, with better cooling to reduce the risk of overheating. There is nothing stopping you from using a standard servo, and running it above it's rated speed, to drive a spindle. Most servo manufacturers will list a rated speed, but they'll also produce speed/torque graphs that show motors performance above the rated speed, up to the speed they deem possible to run the servo.


In terms of drives.
Torque mode is the most basic, and gives the most responsive control over the servo. The big downside is torque mode is inherently unstable. Anytime the load on the servo changes, the servo speed also changes, so the controller has to be very responsive and tuned very well to maintain position.

Speed mode gives almost as much control over the servo, but due to the additional filtering involved, it's not quite as responsive. Speed mode was the mode any old school DC servo with a tacho fitted used, as at the time, controllers just weren't responsive enough to use torque mode. The tacho and servo drive (well technically amplifier) essentially acted as a damper to reduce the responsiveness needed from the controller to maintain position.

Position mode sits on top of speed and torque modes. It's basically a closed loop controller within the servo drive.

How any servo and drive responds, is entirely down to settings.
I'll just correct Jazz on torque mode. If you set a servo to torque mode, and command it to produce say 50% torque. With no load it'll spin up to it's rated speed. Stall it, it'll sit and produce 50% torque against that stall. Load it so it spins the opposite way, and it'll still happily sit producing that 50% torque against how it's being spun.
The servo drive doesn't care how the motor is turning/being turned, as long as it can produce the request torque without triggering a fault, it'll continue to do so.

Servo tuning does vary between axis and spindle though.
In an axis you want positional accuracy, so they get tuned to hold that position as best as possible. The downside is you end up working the motor harder, as it continually changes output trying to hold position, and you'll often find what works well at low speed/stationary, doesn't work so well at high speed, so you need to compromise to minimise following error under all conditions.
With a spindle, you're not as concerned about position, so you can tune them to be far more sluggish to respond to changes. A spindle being a few turns from position at 3000rpm isn't a major issue, as long as the speed remains constant. The last thing you want is a spindle that surges when it encounters a sudden load change, so you'll generally tune them to be more sluggish to respond.
As somebody asked me when tuning the servo spindle on my little mill, why do you need a spindle that only has a 20 count position error?
Even when rigid tapping, all you need is a stable spindle that you can reliably stop, it doesn't have to be on position to achieve that. It's up to the Z axis is ensure the thread is where it should be.

Although I still love playing with the servo spindle on my lathe, but it highlights how sluggish it's tuned. Put it in C-axis mode, and you can rock the chuck, before the drive just sluggishly pulls it back to where it should be. It could probably be tuned to respond faster, but it wouldn't make any difference to the parts it's makes.
You have to consider the loads the spindle actually sees. My lathe has a 5.5KW 3000rpm spindle servo, so it only produces 17Nm of continuous torque at the chuck. Given it's got a 8" chuck fitted, that means it only takes 170N pull at the edge of the chuck to overcome the spindle. Doesn't sound a lot, but when you're drilling/milling within the capacity of the lathe, it's more than enough to keep things where they need to be.