Power Screw Raising Torque (Square Thread)

A power screw is an inclined plane wrapped round a cylinder, and the torque to drive one is the same problem as pushing a block up a ramp against friction — unrolled, the thread is exactly that ramp. A screw jack, a machine vise, a press ram, a lathe cross-slide and a linear actuator are all the same element with different clothes on.

Unroll one turn of the thread and you get a right triangle: base $\pi d_m$, the mean circumference; height l, the lead, which is the axial travel in one turn. The angle between them is the lead angle, $\lambda = \arctan(l/\pi d_m)$. Push a block up that ramp against a friction coefficient µ and the required tangential force is $F(\tan\lambda + \mu)/(1 - \mu\tan\lambda)$; multiply by the radius $d_m/2$ to get torque, and multiply top and bottom by $\pi d_m$ to clear the tangents, and the bracket on this page falls out exactly as written.

Work an example. A 25 mm mean diameter single-start screw with a 5 mm lead, µ = 0.08, raising 10 kN. The numerator is $0.005 + \pi(0.08)(0.025) = 0.01128$; the denominator is $\pi(0.025) - 0.08(0.005) = 0.07814$; their ratio is 0.1444; and $T = (10000 \times 0.025/2)(0.1444) = 18.05$ N·m. Now notice how little of that work reaches the load. One turn raises 10 kN by 5 mm — 50 J — and takes $2\pi T = 113$ J of input. The efficiency is 44 per cent, and the missing 63 J went into rubbing the flank.

That inefficiency is not a defect; on a jack or a vise it is the product. A screw is self-locking when the friction angle exceeds the lead angle, and a self-locking screw can never be more than 50 per cent efficient in the raising direction. You are buying the property that the load stays up when the handle is released, and you pay for it in energy. Where you would rather have the efficiency — a fast actuator, a machine-tool feed — you use a ball screw at 90 per cent and accept that it back-drives, and you fit a brake.

Two things this page deliberately leaves out, both of which matter. It is the SQUARE-thread form: an Acme thread's flank is tilted 14.5° from the radial and an ISO trapezoidal thread's 15°, which raises the normal force and therefore the friction by about 3.5 per cent, and the standard fix is to use µ/cos(α/2) in place of µ. And it is the THREAD torque only — a screw turning against a plain thrust collar adds $F\mu_cd_c/2$, which on a jack is frequently as large again as the whole figure above, and which a rolling thrust bearing all but eliminates. That bearing is usually the cheapest efficiency available on a screw.

Finally, sizing a screw is not this calculation alone. The load it can raise is also limited by column buckling if it is long and slender in compression, by the bearing pressure on the thread flank, which sets how many turns of nut engagement are needed and how fast it wears, and by the shear area at the thread root. And µ itself is the weakest number in the whole exercise — 0.10 to 0.20 for lubricated steel on steel, 0.06 to 0.10 for steel on bronze, far worse dry or dirty, and different again once the screw has run in. Two significant figures on the torque is optimistic.