Stepper motor and Ball Screw torque calculation

I am trying to calculate the nema23 motor needed for my Z axis.

My Z axis with spindle has 22lbs and chinese ball screw SFU1610 (diameter 16mm / pitch (lead) 10mm)

After much searching for discussions and articles, I found just that
https://buildyourcnc.com/FAQ/879

https://i.stack.imgur.com/8qpvW.png

Is this formula correct? Does it apply to the use of ball screw as well (example is with lead screw)?

Effort = Sf + (Load/(2 x pi x (R/p) x Se))

Load = 97 N (22 lbs)
R = 0.314961 inch (8 mm)
p = 0.393701 inch (10 mm)


Effort = Sf + (Load/(2 x pi x (R/p) x Se))


Effort = 5 N + (97 N / (2 x 3.14 x (.314961 / .393701) x .2))


Effort = 5 N + (97 N / (6.28 x .8 x.2))


Effort = 5 N + (97 N / 1)


Effort = 5 N + 97 N


Effort = 102 N = 22 lbs = 352 oz-in

Hi Vitali- Heres a simple approach. Plus remember that if you need more acceleration it will require more torque. Peter

Quote:

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Originally Posted by peteeng

Hi Vitali- Heres a simple approach. Plus remember that if you need more acceleration it will require more torque. Peter

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Thanks Peter!

I will study more about that

Very useful thank you

Hi,
the acceleration calculation goes like this:

dw/dt= T / J_total

where J_total is the first moment of inertia of all components, both rotating and linear.
w is angular velocity in radians/s
and T is the applied torque in Nm.

The calculation is really about finding the first moment of inertia.

J_total = J_armature + J_ballscrew + J_{linear axis}

J_armature is easy, just read the manufacturers spec. For example:
Object reference not set to an instance of an object.
This is a typical medium/high torque 23 size stepper and has a J=0.84 x 10^-4 kg.m².

You don't say how long your ballscrew is, I assume 300mm:

J_ballscrew =m . R²/2
where m is the mass of the ballscrew, R is the radius of the ballscrew
m(mass) =l x pi R² x rho where rho is the density of steel, 8000kg/m³
=0.3 x 0.008²x PI x 8000
=0.482 kg
therefore:
J_ballscrew= 0.482 x 0.008²/2
=0.15 x 10^-4 kg.m²

J_{linear axis}=m . (pitch/2 PI)²
where m is the axis mass (22lb=10kg), pitch is the ballscrew pitch:
J_{linear axis}=10 x (0.01/2.PI)²
=10 x 0.00159²
=0.253 x 10^-4 kg.m²

We now combine the individual moments:
J_total= (0.84 + 0.15 + 0.25) x 10^-4 kg.m²
=1.24 x 10^-4 kg.m²

Firstly note that the dominant term in the total momentum is actually the armature of the stepper (0.84) followed by the linear axis (0.25), at about 1/3 that of the
stepper alone! The ballscrew is the smallest but not insignificacnt term (0.15). Note also that the first moment of inertia of the ballscrew depends on mass, ie proportional
to the square of the radius and also the radius of gyration, again proportional to the square of ballscrew radius. The net result is that the ballscrew moment is dependent
on the ballscrew radius to the fourth power. Ballscrew radius ( equivalently diameter) is critical in this calculation.

Lets now calculate the angular acceleration of such a system with 1Nm torque applied:
dw/dt= 1 / 1.24 x 10^-4
=8064 rad/s²

and given that the axis velocity is related to angular velocity:
v= pitch x w/2 .PI so
dv/dt = pitch x dw/dt/2.PI
= 0.01 x 8064/6.282
=12m/s² or 1.2g which is amazing.

1Nm =192 oz.in. Remember that steppers lose torque bigtime the faster they go so if you choose low inductance steppers then you might expect to retain 60% of the torque
at 500 rpm then 192/ 0.6=320 oz.in should be enough torque to accelerate your axis at 1.2 g. Note that if you want thrust in addition to acceleration then you need more torque,
so 500oz.in is looking like your best bet.

Craig

Quote:

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Originally Posted by joeavaerage

Hi,
the acceleration calculation goes like this:

dw/dt= T / J_total

where J_total is the first moment of inertia of all components, both rotating and linear.
w is angular velocity in radians/s
and T is the applied torque in Nm.

The calculation is really about finding the first moment of inertia.

J_total = J_armature + J_ballscrew + J_{linear axis}

J_armature is easy, just read the manufacturers spec. For example:
[url=https://www.leadshineusa.com/productdetail.aspx?type=products&category=stepper-products&producttype=stepper-motors&series=cm&model=d57cm31[/url] Object reference not set to an instance of an object.[/url webpage
This is a typical medium/high torque 23 size stepper and has a J=0.84 x 10^-4 kg.m².

You don't say how long your ballscrew is, I assume 300mm:

J_ballscrew =m . R²/2
where m is the mass of the ballscrew, R is the radius of the ballscrew
m(mass) =l x pi R² x rho where rho is the density of steel, 8000kg/m³
=0.3 x 0.008²x PI x 8000
=0.482 kg
therefore:
J_ballscrew= 0.482 x 0.008²/2
=0.15 x 10^-4 kg.m²

J_{linear axis}=m . (pitch/2 PI)²
where m is the axis mass (22lb=10kg), pitch is the ballscrew pitch:
J_{linear axis}=10 x (0.01/2.PI)²
=10 x 0.00159²
=0.253 x 10^-4 kg.m²

We now combine the individual moments:
J_total= (0.84 + 0.15 + 0.25) x 10^-4 kg.m²
=1.24 x 10^-4 kg.m²

Firstly note that the dominant term in the total momentum is actually the armature of the stepper (0.84) followed by the linear axis (0.25), at about 1/3 that of the
stepper alone! The ballscrew is the smallest but not insignificacnt term (0.15). Note also that the first moment of inertia of the ballscrew depends on mass, ie proportional
to the square of the radius and also the radius of gyration, again proportional to the square of ballscrew radius. The net result is that the ballscrew moment is dependent
on the ballscrew radius to the fourth power. Ballscrew radius ( equivalently diameter) is critical in this calculation.

Lets now calculate the angular acceleration of such a system with 1Nm torque applied:
dw/dt= 1 / 1.24 x 10^-4
=8064 rad/s²

and given that the axis velocity is related to angular velocity:
v= pitch x w/2 .PI so
dv/dt = pitch x dw/dt/2.PI
= 0.01 x 8064/6.282
=12m/s² or 1.2g which is amazing.

1Nm =192 oz.in. Remember that steppers lose torque bigtime the faster they go so if you choose low inductance steppers then you might expect to retain 60% of the torque
at 500 rpm then 192/ 0.6=320 oz.in should be enough torque to accelerate your axis at 1.2 g. Note that if you want thrust in addition to acceleration then you need more torque,
so 500oz.in is looking like your best bet.

Craig

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Thanks a lot!