What the heck. I’ll throw my guess in as well. Assuming the belt pitch is 2mm, I would say it’s (36 teeth/rev / 90 teeth * 36 teeth * 2mm/tooth = 28.8 mm/rev). This would be .144 mm/step at 1x microstepping, .018 mm/step at 8x, and .009 mm/step at 16x. This would result in 6.9444 steps/mm at 1x, 55.5555 steps/mm at 8x, and 111.1111 steps/mm at 16x.
Unless I’m misunderstanding how the pulleys and belts are actually configured, I don’t believe the diameter (or circumference) of the m8 shaft factors in at all.
With the 112.1 number you gave, I assume that you’re running 16x microsteps and just happen to be close to the correct number of 111.111…
If what I typed above is the case here is your steps per mm for the gt2 belt connected to the bed.
((360deg/rotation)/(1.8deg/step))*(16microstep/step) = 3200 microstep/rotation of the stepper
Now gear reduction:
36tooth/90tooth = 0.4 ratio motor to shaft
Output is a 36 tooth pulley so one motor rotation is:
(36tooth/rotation)(2mm/tooth)(0.4 ratio)= 28.8mm travel per rotation at the stepper
So (3200microsteps/rotation)/(28.8mm/rotation)= 111.11111… microsteps/mm
It would be better to get this to be an integer number to avoid rounding errors.
Someone should double check my work because I did this in my head while driving.