I like how to calculate the step for Mach when using step motors. I plan to build another new machine using servo motor and gear rack.

The equipment would be like this.

Servo Motor RPM = 3200
Gear Rack = 20 DP
Pinion = 20 Teeth
GeckDrive = 320
Decoder = 500 CPR
Gearbox/belt = 9

I think that is everything I need. How do I get the steps for Mach?

What are you meaning by the Gearbox/belt = 9? is that a reduction of 9?
Without any gearing then 1 rev of motor moves 1 inch, one rev of motor is 2000 pulses (500 x 4) so that is what your steps per unit will be.
 If you have a 9:1 gearing then it will need 9 revs to move 1 inch so Steps per would be 2000 x 9 = 18,000.
Hood

Thank you for the clear explanation.

What are you meaning by the Gearbox/belt = 9? is that a reduction of 9?
Without any gearing then 1 rev of motor moves 1 inch, one rev of motor is 2000 pulses (500 x 4) so that is what your steps per unit will be.
 If you have a 9:1 gearing then it will need 9 revs to move 1 inch so Steps per would be 2000 x 9 = 18,000.
Hood

in this calculation, how many step per do we need to turn this servo 3200 RPM ?

Depends what  line count your encoder is.
Hood

if we have 2000 line encoder,, how many step per do we need to turn this same servo motor 3200 RPM

If its a 2000 line encoder then that is 8,000 in quadrature so 8,000 x 3,200 = 25,600,000 pulses per min/60 = 426,666 pulses per second (426KHz)
If it was a 500 line encoder then it would be 2,000 x 3,200 / 60 = 106,666Hz (106KHz)

Hood

i understood that i'll need 426Khz for 2000 line - 8000 in quadrature for 3200 RPM ! but what will be the "step per" in this calculation ?  ( 8000 ? )

It all depends on what the pitch of the ballscrews are and if you have any reduction between the motor and the ballscrews.
Hood

i didnt mean any movement of axis.. i just mean, how many  "step per" do we need to turn motor 3200 RPM with 2000 line encoder.. we need totally 426,666 KHZ but.. what could we set "step per" from motor tuning to turn servo 3200 RPM ??