Power Screw Torque (raise)⚠ unverified

Understanding the Parameters

• Axial load $F$ — the load being lifted; torque is directly proportional to it.

• Mean diameter $d_m$ — appears both as the moment arm ($d_m/2$) and in the friction geometry. A larger screw needs more torque for the same load but distributes thread stress over more area.

• Lead $l$ — the "useful" advance per turn. A larger lead lifts the load faster (higher efficiency) but demands more torque and erodes self-locking. It sets the numerator's first term — the ideal, frictionless lifting torque.

• Friction coefficient $\mu$ — governs the friction terms. For power screws $\mu \approx 0.1$–$0.2$ (steel on steel/bronze). Friction dominates: typically well over half the raising torque is spent on it, and it is what makes the screw self-locking.

• Thread half-angle $\alpha$ — the flank angle. A square thread ($\alpha = 0$, $\sec\alpha = 1$) is the most efficient because the load normal force is purely axial. An Acme thread ($\alpha \approx 14.5^\circ$) is easier to manufacture and adjust for wear but its $\sec\alpha \approx 1.03$ raises friction slightly.

Derivation (Approaching a Proof)

Unwrap one turn of the thread at the mean diameter into an inclined plane of lead angle $\lambda$, where $\tan\lambda = l/(\pi d_m)$ (see Screw Lead Angle). Raising the load is pushing the weight $F$ up this plane with a horizontal force $P$ applied at radius $d_m/2$; the torque is $T = P\,d_m/2$.

For a square thread, force balance up the incline against Coulomb friction (normal force $N$, friction $\mu N$) gives the classic inclined-plane result:

$$P = F\,\frac{\tan\lambda + \mu}{1 - \mu\tan\lambda}.$$

Substitute $\tan\lambda = l/(\pi d_m)$ and multiply numerator and denominator by $\pi d_m$:

$$P = F\,\frac{l + \pi\mu d_m}{\pi d_m - \mu l} \;\Longrightarrow\; T_R = \frac{P d_m}{2} = \frac{F d_m}{2}\,\frac{l + \pi\mu d_m}{\pi d_m - \mu l}.$$

For a thread with flank angle $\alpha$ (Acme, V), the flanks are inclined, so the normal force is larger by $1/\cos\alpha = \sec\alpha$; the effective friction becomes $\mu\sec\alpha$. Replacing $\mu$ with $\mu\sec\alpha$ in the friction terms yields the general form:

$$T_R = \frac{F d_m}{2}\,\frac{l + \pi\mu d_m\sec\alpha}{\pi d_m - \mu l\sec\alpha}.$$

The bracket exceeds $l/(\pi d_m)$ whenever friction is present, so a real screw always needs more than the ideal frictionless torque $F l/(2\pi)$ — the gap is the efficiency loss (Power Screw Efficiency). A practical screw also adds collar friction (Collar Torque) for the total (Total Torque Raise).

Dimensional check. The bracket is (length)/(length) = dimensionless; $[T_R] = \text{N}\cdot\text{m}$. ✓

History and Development

Power-screw torque analysis is a classic application of the inclined-plane and Coulomb friction, dating to the machine-design foundations of the 19th century and standardised in Shigley. Square threads are preferred for maximum efficiency in power transmission; Acme threads trade a little efficiency for manufacturability and a split-nut/wear-adjustment capability. This equation with the $\sec\alpha$ generalisation is identical to Acme Thread Torque (which folds $\mu\sec\alpha$ into an effective friction $\mu_{\text{eff}}$).

Related Concepts: Torque Lower Load, Acme Thread Torque, Collar Torque, Total Torque Raise, Power Screw Efficiency, Screw Lead Angle, Bolt Torque Preload

Notes: Set $\alpha = 0$ for a square thread ($\sec\alpha = 1$). Self-locking when $\mu\sec\alpha > \tan\lambda$. Add Collar Torque for the total driving torque. Same formula as Acme Thread Torque via $\mu_{\text{eff}} = \mu\sec\alpha$.