# Static analysis of a lead screw and disc friction

I'm doing a static analysis of a square thread lead screw. I think this site gives a good description of the problem.

$$M = W \space R \space tan(phi_s + alpha)$$

Where M is the moment required to raise the screw to impending motion, W is the force load on the screw, R is the mean radius of the screw, phi_s is the screw friction angle, and alpha is the screw thread pitch.

Further down on the same site a disc friction problem is looked at.

Disc Friction Statics

Here the moment is calculated by integrating with respect to the radius.

Adapting this approach to analyze the screw thread, the moment required to raise the screw would be.

$$M = \frac{2}{3} \frac{(R_o^3 - R_i^3)}{(R_o^2 - R_i^2)} W \space tan(phi_s + alpha)$$

Where R_o is the outside radius and R_i is the inside radius.

These yield similar but different results. My question is, is the first approach using the mean radius simply a simplification/approximation of the disc friction method of integrating over radius and the later will be more accurate? Or is it incorrect to integrate over the radius when looking at a screw thread analysis? If so, can you please elaborate on why.

• Using the mean radius or effective radius looks like a simplification. All Integrating does is take into account the fact that the same local piece of frictional force (p.dA) contributes more moment when it's out at a further radius. In the case of a power screw, $r_o/r_i$ isn't too far from 1, so the error from this approximation would be modest. And the overall error would anyway be dominated by uncertainty in the friction coefficient. Commented Feb 12 at 20:20