Acme Thread Torque⚠ unverified
Mechanical / Power Screws · Compute the raising torque for an Acme (modified square) thread
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| F | F | N | 1.0 | Axial load on the screw |
| dm | dm | m | 1.0 | Mean (pitch) diameter of the thread |
| l | l | m | 1.0 | Lead of the thread (axial advance per revolution) |
| mu | μ | — | 1.0 | Coefficient of friction between the screw and nut threads (dimensionless) |
| alpha | α | — | 1.0 | Thread half-angle, in degrees |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | T | N*m | Raising torque for the Acme thread, in newton-metres (N*m) |
The science & history
Understanding the Parameters
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Friction $\mu$ and half-angle $\alpha$ — combine into $\mu_{\text{eff}} = \mu/\cos\alpha$. The flank angle tilts the thread contact so the normal force (and hence friction) is larger than for a square thread by $\sec\alpha$. For Acme, $\cos 14.5^\circ \approx 0.968$, so $\mu_{\text{eff}} \approx 1.03\,\mu$ — only a few percent friction penalty over a square thread, which is why Acme is such a popular compromise.
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The other parameters ($F$, $d_m$, $l$) — behave exactly as in the general raising-torque formula: load and moment arm scale the torque, lead sets the useful lift per turn.
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Why an angled thread at all? The flanks let a split nut close on the screw and take up wear (you cannot easily adjust a square thread for wear), and the tapered root is stronger and easier to cut than a square groove — practical advantages worth the small $\sec\alpha$ friction cost.
Derivation (Approaching a Proof)
The derivation is identical to the general power-screw raising torque (Power Screw Torque raise) up to how the flank angle enters. On the unwrapped inclined plane, the load $F$ presses normal to the thread surface. For a square thread that normal is purely radial/axial; for an Acme thread the flanks are inclined at the half-angle $\alpha$, so the actual normal force on the flank is
$$N = \frac{F}{\cos\alpha} = F\sec\alpha,$$
larger than the axial load because it is spread onto the tilted faces. The friction force is $\mu N = \mu F\sec\alpha$, i.e. the friction behaves as if the coefficient were $\mu_{\text{eff}} = \mu\sec\alpha$. Substituting this effective friction into the square-thread raising torque
$$T = \frac{F d_m}{2}\,\frac{l + \pi\mu_{\text{eff}} d_m}{\pi d_m - \mu_{\text{eff}} l}$$
gives the Acme form. Expanding $\mu_{\text{eff}} = \mu\sec\alpha$ recovers exactly the $\sec\alpha$-in-each- term version of Power Screw Torque raise — confirming they are one equation.
Dimensional check. The bracket is dimensionless; $[T] = \text{N}\cdot\text{m}$. ✓
History and Development
The Acme thread form was standardised in the USA in 1895 to replace the harder-to-make square thread while retaining most of its power-transmission efficiency; the 29° angle allows a split nut for wear compensation. The effective-friction treatment ($\mu\sec\alpha$) is the standard way Shigley and machine- design texts handle angled power-screw threads.
Related Concepts: Power Screw Torque raise, Torque Lower Load, Collar Torque, Total Torque Raise, Power Screw Efficiency, Screw Lead Angle
Notes: Same equation as Power Screw Torque raise with $\mu_{\text{eff}} = \mu\sec\alpha$. For a square thread $\alpha = 0$ so $\mu_{\text{eff}} = \mu$. Add Collar Torque for the total driving torque.