Power-Screw and Lead-Screw Torque and Efficiency

Three geometric parameters drive the torque equations. The mean thread diameter d_m = d − h sets the effective friction radius r_m = d_m / 2. The lead l (distance advanced per revolution; l = n_starts × pitch) determines the helix angle λ. The thread flank angle α controls how much of the normal force between mating threads acts radially — creating an extra wedging component that raises the effective friction.

For a flat (square) thread, α = 0 and the thread flanks are perpendicular to the screw axis, so the full normal force opposes axial motion and no radial wedging occurs. For an Acme thread (ASME B1.5, α = 14.5°) or a metric trapezoidal thread (DIN 103, α = 15°), the flanks are inclined and the radial component increases the contact force. The engine corrects for this by replacing the measured friction coefficient μ with a modified value μ′ = μ / cos(α). Buttress threads (α ≈ 7° on the load flank) fall between the two extremes. The self-locking condition and both torque expressions use μ′ throughout.

The raise torque has two additive parts: the thread torque T_thread (which does the work of lifting the load against thread friction) and the collar torque T_c (which overcomes friction at the thrust bearing or flat collar that reacts the axial load). The thread torque follows from the equilibrium of forces on the thread helix, expressed at the mean radius r_m. The collar torque is simply the product of the collar friction coefficient, the load, and the mean collar radius.

The lower torque expression is similar but with the sense of the friction term reversed. When the load attempts to drive the screw backwards, friction now partially assists the motion. If the load can overcome friction — that is, if tan(λ) > μ′ — the screw overhauls (back-drives) and a separate brake or holding device is required. When tan(λ) ≤ μ′, friction is sufficient to prevent back-driving and the screw is self-locking: a negative T_lower_thread confirms the load cannot move the screw unaided. The collar torque is always added in the same (positive) direction for both raising and lowering because the collar must always be overcome by the drive.

Thread efficiency η compares the ideal (frictionless) work of raising the load over one revolution — which is simply W × l — to the actual thread-raising work 2π × T_raise_thread. It excludes collar losses because the collar is a separate mechanical component (thrust bearing, plain collar or washer) whose efficiency depends on its own design, not on the thread geometry. Collar losses are tracked separately in the total torque and in the mechanical-advantage panel.

Efficiency rises with increasing lead angle up to a peak near λ_opt ≈ 45° − α/2, then falls again because very steep leads require large normal forces for the same axial load. At the self-locking boundary, η → 0; at the optimal lead angle, η_max ≈ (1 − μ′) / (1 + μ′) for small α. In practice, most power screws operate between 10° and 20° lead angle, putting thread efficiency in the 20%–45% range for typical friction coefficients. The MechanixCalc power screw calculator plots η against λ for five standard friction levels so the designer can see the sensitivity.

The screw minor (root) diameter d_r = d − 2h carries the combined compressive and torsional loads. Compressive stress σ_c acts on the cross-sectional area π·d_r²/4; torsional shear stress τ acts on the same section from the raising thread torque. Von Mises theory combines them into a single equivalent stress σ_vm for comparison against the material yield strength. This check is particularly important for long, slender screws where d_r is small relative to the applied load.

Thread contact (bearing) pressure p_c acts on the projected flank area of the engaged threads. For a nut of engagement length L_nut, the number of engaged threads is n_t = L_nut / l, and the bearing pressure is W spread over π·d_m·h per thread. The Shigley limit is 15 MPa for a general nut and 25 MPa for a bronze nut — values exceeded at low n_t or high load indicate a longer nut or coarser thread is needed to avoid galling and accelerated wear.