Lead Screw & Ball Screw Torque Calculator

How it works

Sliding screws: T_raise = F (d_m/2) × (l + π µ d_m sec α) ÷ (π d_m − µ l sec α), T_lower = F (d_m/2) × (π µ d_m sec α − l) ÷ (π d_m + µ l sec α), α = 14.5° ACME, 15° Tr, 0° square · collar T_c = F µ_c d_c ÷ 2 · efficiency = F l ÷ (2π T_raise) · self-locking if µ > tan λ × cos α · ball screw: T = F l ÷ (2π η) · P = T ω · T_acc = J ω ÷ t · n_crit = λ × 10⁷ × d_r ÷ L² · P_buckle = η × 10⁴ × d_r⁴ ÷ L²

A screw thread is an inclined plane wrapped around a cylinder, and turning it pushes the load up that plane against friction. The torque formula is the force balance on a block sliding up a ramp whose slope is the lead angle; the secant of the thread half-angle accounts for the wedging of a V-shaped ACME (29°) or metric trapezoidal (30°) thread. A thrust collar or bearing that carries the axial load adds its own friction torque. Fine leads give large force from small torque but at poor efficiency, and when friction exceeds the tangent of the lead angle the load cannot turn the screw backwards — self-locking, which is why a jack holds a car with the handle removed. Ball screws replace sliding with rolling balls, lifting efficiency to around 90 percent at the price of back-driving, so vertical axes on ball screws always carry a brake. To size a motor, add the torque needed to spin up the screw and the moving mass in the acceleration time, then a margin for breakaway friction. Long, fast screws have two more limits: the critical speed, where the screw starts to whip like a skipping rope, and the buckling load in compression; both fall with the square of the unsupported length and depend on how the ends are held. The critical-speed and buckling constants used here are the usual catalogue ones and already include safety factors of 0.8 and 0.5.

Worked example

Raising 5,000 N with a Tr20×4 screw treated as ACME (mean diameter 18 mm, µ 0.15) takes 10.27 N·m at the thread plus 0.09 N·m in a rolling thrust bearing — 10.36 N·m (91.7 lbf·in) in total at 31% efficiency. The screw is self-locking and needs 3.84 N·m to lower the load. At 300 rpm the load moves at 20 mm/s and needs about 325 W; with a 1.5× margin the motor should give about 15.5 N·m. A 500 mm screw held fixed–supported, with a root diameter of about 15 mm, has a critical speed near 9,000 rpm and an allowable buckling load of about 20,000 N — four times the load — so both are fine. A 16 mm ball screw with a 5 mm lead would need only 4.4 N·m but would back-drive.

Sources: Engineering ToolBox — Screw jacks: effort force and torque

Frequently asked questions