Converting Between Linear Force and Torque of a Ballscrew

Force = Torque x 2 x PI x gear reduction ratio x %eff / Lead

Torque = Force x Lead x %eff / (2 x PI x gear reduction ratio)

Derivation:

If you know the torque applied to a ballscrew, then you can calculate the linear force produced by the ballscrew.

If you know the required linear force of a ballscrew, then you can calculate the torque that must be applied.

If you know the linear force applied to a ballscrew, then you can calculate the torque produced by backdriving the ballscrew.

The calculation is based on converting the applied torque to the force acting on the ball bearing, and then calculating the linear vector component of the resultant force.

Torque = Force x Distance

= (Rotational Force acting on a ball bearing) x (radius of ballscrew)

So you can calculate the rotational vector component of force based on torque and the radius of the ballscrew.

T = F1 x D/2, where:

F1 is the vector component (tangent to the ballscrew) of the force applied by the screw onto the ball bearing.

D is the diameter of the ballscrew

So, F1 = 2T / D

Linear force is produced because of the angle of the ballscrew thread.  The angle is specified by the lead, distance per revolution (distance per circumference).

F1 is the vector component in the direction of rotation of the ballscrew.  It is proportional to the lead of the ballscrew.  The linear force is the vector component in the axial direction of the ballscrew, and it is proportional to the circumference of the ballscrew.

So F2 / F1 = PI x D / L, where:

F2 is the linear vector component of force

L is the lead, distance per rev

Solving for F2, we have

F2 = F1 x PI x D /L = T x 2 x PI x D / (L x D)

F2 = T x 2 x PI / L

The units can be in SI or English: in-lbs, lbs, and inches, or N-m, N, and meters.