Z Axis Motor selection

The force of gravity multiplied by the mass of the Z-Axis is equal to downwards force. The motor needs to be able to hold this and lift this on a 20mm (Minor diameter: 16.5) screw with a 5mm pitch. If the force of cutting is greater than the force generated by the weight of the gantry and the need to accelerate the gantry in an upwards then this will be used in the calculations.

Force generated by Z axis = 37.12KG (2.D.P) * 9.81 = 364.1472N


http://www.esc-ltd.co.uk/Drilling%20Formulas.pdf

Fr=0.63* ((0.305*10*500)/2)= 480.375

480.375-364.1472 = 116.2278

As force is acting down upon the cutting tool from the relationship of the gantry and gravity.


L= 37.12*(2*9.81) = 728.2944N (Note: Two times gravity to allow equal acceleration in both Z directions)

Angle of  lifting plane = α
= tan^-1(Pitch/Screw minor diameter*π)
= tan^-1(5/16.5*π)
= 5.51 degree (2.D.P)

μ = 0.01 (http://www.kssballscrew.com/us/pdf/qa/Q-BS-12.pdf)

Friction Angle = β
 = tan^-1 * μ
 = tan^-1 * 0.01
= 0.57 degrees(2.D.P)

F =L tan (β + α)

   = 728.2944 * tan (0.57+5.51)
   = 77.56N (2.D.P)
  
T = F x radius
   = 77.56*0.00825 = 0.63987 Nm

The motor will require a holding force of at least 0.63987 Nm

http://www.ebay.co.uk/itm/DMM-AC-Servo-Motor-Drive-4Axis-750W-for-CNC-Router-Plasma-Mill-kit-/161346672390?pt=LH_DefaultDomain_2&hash=item259102d706

If none of the other servo motors required within the machine require more than 2nm continuous torque this should provide a good solution as it incorporates other required components.