steps per mm for belt driven leadscrews

Just as the title says really, I'm looking to drive my z axis with a belt. I've searched around and all that google / the web shows up is the josef prusa calculators, which is great but says nothing about leadscrews+belts. So what is the calculation for figuring out the steps per mm?

A bit of of info on my set up -

1.8 deg nema 17 (59Ncm I think, so should be beefy enough)
16x microstepping
1140 tooth closed loop gt2 belt
2x tr8x8 lead screw, 4 start. 8mm travel per revolution.
32 tooth pulley on the nema 17
20 tooth pulley on the leadscrew

The pulleys are what I had lying around, although I'm not sure what an effective ratio would be here? Any suggestions on this would be great as well

Edited 1 time(s). Last edit at 06/23/2016 06:12AM by Origamib.

It takes 16 usteps/step x 200 steps/rev = 3200 usteps/rev.
The 32 tooth pulley turns once when the motor turns once, and since the pitch is 2 mm, the belt moves 2 mm/tooth x 32 teeth/rev = 64 mm/rev
The screw has a 20 tooth pulley. When the motor turns one rev, the screw turn 32 tooth/rev / 20 tooth/rev = 1.6 revs.
The screw lead is 8 mm/rev. When the motor turns one rev, the screw turns 1.6 revs so the Z axis moves 1.6 rev x 8 mm/rev= 12.8 mm
It took 3200 usteps (1 motor rev) to move the Z axis 12.8 mm, so 3200 usteps/rev / 12.8 mm/rev = 250 usteps / mm

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Ultra MegaMax Dominator 3D printer: [drmrehorst.blogspot.com]

The Prusa calculator still works, even if you're driving the screw with a belt.
It's all about the ratio of the pulleys.
Since you're saying that you're running 32 teeth on the stepper and 20 on the screw, you're running 1.6 : 1 gear ratio, anything else doesn't matter about the belt system.
Since that calculator doesn't do floats in the inputs for the ratio, it's easiest to put in 16:10 ratio unless you want to find out the lowest integer ratio equal to yours (though it isn't that hard, it's 8:5).

Edited 1 time(s). Last edit at 06/23/2016 02:01PM by Mikk36.

While on-line calculators are nice, there's real value in thinking through the problem yourself. When you use an on-line calculator you're trusting the person who wrote it to have done it correctly. If you can't figure out the answer for yourself, how can you know if the on-line calculator is correct? Simple dimensional analysis works for this and almost any other 3D printing related calculation.

The OP asked what is the gear ratio of a 32 tooth pulley driving a 20 tooth pulley. The word "ratio" tells you what to do: 32/20 is the ratio, by definition.

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Ultra MegaMax Dominator 3D printer: [drmrehorst.blogspot.com]

Maths is not my strong point unfortunately... Coincidentally, I had almost figured it out through weird backwards math in my head... Your method was much more straightforward though!

Another way to calculate it - ignoring micro steps makes it easier (and you did say you wanted steps per mm, not micro steps per mm).

So 200 steps = 1 revolution (360degress / 1.8 degrees per step). So with 1:1 gearing 200 steps = thread pitch (in this case 8mm). So 1mm is 200/8 = 25 steps. But your gearing is 32:20 so the steps per mm are 25 *20 / 32 = 15.625. If you multiply that by 16 you'll get the 250 micro steps that DD came up with.

Personally I don't like to rely on micro stepping for positional accuracy (just something I read somewhere). But if you simply switch the pulleys round you get 25*32/20 = 40 steps per mm which I think is a much nicer number than 15.625. You'll also increase the torque but lose a bit of speed but that maybe is better for the Z axis.

Juts my twopence worth

How about
with known Steps / mm set in firmware
Command Z axis movement A
Measure the Z Axis distance moved B

A little calculation
Reset steps/mm
until
commanded distance = distance moved

Record result --- so you can put it back when the EEPROM loses it.

Quote
the_digital_dentist
The OP asked what is the gear ratio of a 32 tooth pulley driving a 20 tooth pulley. The word "ratio" tells you what to do: 32/20 is the ratio, by definition.

Well, yeah, guess I was also overthinking it.

Quote
cozmicray
How about
with known Steps / mm set in firmware
Command Z axis movement A
Measure the Z Axis distance moved B

A little calculation
Reset steps/mm
until
commanded distance = distance moved

Record result --- so you can put it back when the EEPROM loses it.

That's a bit stupid way of solving it, if you have known constants of the parts.
You might get even false results from that with bad measurements.

Edited 1 time(s). Last edit at 06/23/2016 04:10PM by Mikk36.

Quote
Mikk36
That's a bit stupid way of solving it, if you have known constants of the parts.
You might get even false results from that with bad measurements.

Actually, using the math gets you close, but if you're concerned about accurate printing, the final calibration has to be done using measurements of prints. I have found -0.5% error (when printing ABS- shrinkage?), so my steps/mm values correct for that (yes, I'm that anal...). That's how you get parts that are the right size on the first attempt.

Edited 1 time(s). Last edit at 06/23/2016 07:39PM by the_digital_dentist.

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Ultra MegaMax Dominator 3D printer: [drmrehorst.blogspot.com]

Shrinkage should not be accounted for in the steps calculation, because it varies by material, heat settings etc.
Calibration of movements should be strictly by mechanical movements. In case of ball screws etc, which are precision machined to match the spec (instead of printed pulleys etc), there is a spec you can assume from them. 8 mm lead will be 8 mm lead, not 7.9 or 8.1 mm.

Edited 1 time(s). Last edit at 06/24/2016 05:43AM by Mikk36.

If you print a part that comes out 0.5% too small, how do you correct it?
Slic3r can't scale in fractional percentages. Cura can.

When I design a part that is 130 mm long I want it to print 130 mm long. I have found that applying the 0.5% to the steps/mm gives me exactly the designed size. I print with ABS 95% of the time, and temperatures are consistently set to 235C and 105C.

Sure I could use Cura and scale the print every time I slice a part, but why would I do that every time I slice, and why would want to restrict myself to using only Cura? Steps/mm is the right place to make the correction because I only have to do it once.

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Ultra MegaMax Dominator 3D printer: [drmrehorst.blogspot.com]