I would like to calculate the maximum safe acceleration for my printer using the following parameters as input:
- printhead weight
- torque at the desired speed
- steps/mm and steps/revolution
Elasticity of frame and belts are ignored.
The Excel file to obtain the torque at the desired speed is available here.
The other parameters are known in advance.
The formula I used for the maximum acceleration is the usual $a=F/m$, where $F=torque*radius$:
$acceleration = torque\ *\ (steps/rev\ /\ steps/mm\ *\ microsteps\ /\ pi\ /\ 2) / mass$
Using 450 g, 0.15 N/m (as predicted for my stepper at 200 mm/s with 24 V), 200 steps/rev, 80 steps/mm, 16 microsteps I obtain about 2100 mm/s2 which seems reasonable and pretty close to standard values.
However, from what I understood the Excel file provides the prediction of a full-step torque, but torque is known to decrease significantly with increasing microsteps.
How to introduce the effect of reduced (incremental) torque for microsteps to calculate a safe maximum acceleration value for a printer?
For information: my stepper and my printer were sold back then with about that acceleration value preset and with that torque at nominal speed (half voltage, half speed than I assumed here).
The print quality was fine, which seems counterintuitive considering the much reduced torque expected at 16X microsteps, about 1/10 of the value I used.
I would expect many lost microsteps with the predefined settings. Or maybe it happens: if all the microsteps are lost, the positioning error would be 16 (micro)steps / 80 steps/mm = 0.2 mm.
As maybe someone noticed, the formula I used
is wrong. The correct one is
Taking this change into account, the final formula is:
$acceleration = torque\ /\ (steps/rev\ /\ steps/mm\ *\ microsteps\ /\ pi\ /\ 2) / mass$
which results, with the given input values, in 52 mm/s2.
This is much less than what is normally set in printers. I doubt that belt stretching and frame flexibility can affect the setting so much, therefore in addition to the microstep aspect, this one should be answered too, since the question is about "theoretically calculate the maximum acceleration".