Torque Lower Load⚠ unverified
Mechanical / Power Screws · Compute the torque required to lower a load on a square-thread power screw
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 | — | 0.0 | Thread half-angle, in degrees. Unused for square threads. Default is 0 |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | T | N*m | Torque required to lower the load, in newton-metres (N*m). A negative value indicates the screw overhauls (load drives the screw) |
The science & history
Understanding the Parameters
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Friction $\mu$ and lead $l$ — the sign of $T_L$ is set by the competition in the numerator between the friction term $\pi\mu d_m$ (which resists motion, favouring self-locking) and the lead term $l$ (the load's tendency to unscrew, favouring overhauling). When $\pi\mu d_m > l$ — equivalently $\mu > l/(\pi d_m) = \tan\lambda$ — friction wins and the screw self-locks.
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The self-locking condition $\mu > \tan\lambda$ — the single most important design criterion for a holding screw. Low-lead-angle screws (small $l$, large $d_m$) with normal friction are self-locking; steep, efficient, or well-lubricated screws may overhaul. This is why a car jack holds but a ball screw (near-frictionless) always needs a brake.
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Thread half-angle $\alpha$ — listed for interface consistency but unused here (this is the square-thread form). For an Acme/V thread, replace $\mu$ with $\mu\sec\alpha$, as in Power Screw Torque raise.
Derivation (Approaching a Proof)
Lowering the load reverses the direction of impending motion on the unwrapped inclined plane (see Screw Lead Angle). Now friction acts up the plane (opposing the descending load), so in the square-thread force balance the signs of the friction terms flip relative to raising:
$$P = F\,\frac{\mu - \tan\lambda}{1 + \mu\tan\lambda}.$$
Substituting $\tan\lambda = l/(\pi d_m)$ and multiplying through by $\pi d_m$:
$$P = F\,\frac{\pi\mu d_m - l}{\pi d_m + \mu l} \;\Longrightarrow\; T_L = \frac{P d_m}{2} = \frac{F d_m}{2}\,\frac{\pi\mu d_m - l}{\pi d_m + \mu l}.$$
The numerator $\pi\mu d_m - l$ carries the physics: if $\mu > \tan\lambda$ it is positive (self-locking, torque needed to release and lower the load); if $\mu < \tan\lambda$ it is negative (the load overhauls, and $|T_L|$ is the braking torque required to prevent runaway). At the boundary $\mu = \tan\lambda$ the lowering torque is exactly zero — the screw is on the verge of self-locking.
Dimensional check. The bracket is dimensionless; $[T_L] = \text{N}\cdot\text{m}$. ✓
History and Development
The lowering-torque sign criterion is the classical statement of screw self-locking, standard in Shigley and machine-design practice. Self-locking makes screws invaluable for holding fixtures, jacks, and clamps; overhauling is deliberately engineered into efficient lead/ball screws that must back-drive (and are paired with brakes). The result descends directly from Coulomb friction on the inclined plane.
Related Concepts: Power Screw Torque raise, Screw Lead Angle, Power Screw Efficiency, Acme Thread Torque, Ball Screw Efficiency
Notes: Square-thread form ($\alpha$ unused; use $\mu\sec\alpha$ for Acme/V). Negative output ⇒ the screw overhauls and needs a brake. Self-locking when $\mu > \tan\lambda = l/(\pi d_m)$.