§1Fastening is not moving
A fastening thread is turned a few times and then holds for years; a power thread runs back and forth under load for its whole life. The two duties want opposite geometry, and the systems page’s flank-angle dial is where they part.
The 60° vee earns its keep by wedging: its steep flanks centre the nut, jam the fit tight and multiply friction — all virtues in a joint that must never move, all vices in one that must move constantly. Run a load up and down a vee thread and the wedging becomes wear, the friction becomes heat, and the 23.2% efficiency the systems page computed becomes a permanent tax on every stroke. The pure answer is the square thread — flanks at 0°, no wedging at all, the efficiency ideal — but the square form is nearly unmanufacturable at scale: its vertical flanks cannot be cut with a simple angled tool or a milling cutter of any sensible geometry, its square root corner is a fatigue notch, and a worn square thread offers no way to take the slack up. The trapezoidal thread is the engineered retreat from that ideal: tilt the flanks just enough to make the form cuttable, strong and adjustable, and no further. How little was given away is exactly what §4’s arithmetic measures.
Contents§2The trapezoidal form
Acme in the inch world, Tr in the metric — near-identical trapezoids of 29° and 30° included angle, half a pitch deep, with generous flats at crest and root.
The proportions are the hero’s: depth p/2, flanks leaning about 15° each side, and crest and root flats around 0.37 p wide. Each feature answers one of the square thread’s failings. The 14–15° flank lets a single straight-sided tool or a disc cutter generate the thread — which is why the thread-milling page ahead treats trapezoidal work as routine — and gives the engaged flanks a slight centring action without meaningful wedging. The wide flat root removes the square form’s corner notch, so a lead screw survives the fatigue duty of a million reversals. And the angled flank makes wear recoverable: as the flanks wear, a nut split along its axis can be closed slightly onto the screw, taking the backlash up — the adjustment every lathe cross-slide nut provides, and the mechanism behind the half-nut that the lathe pages showed clamping onto the leadscrew. Designation follows the metric pattern with a Tr prefix: Tr 30 × 6 is a 30 mm trapezoidal thread of 6 mm pitch; multi-start and left-hand variants are written out exactly as on the metric page, and matter more here, because power screws are where multi-start threads earn their living.
Contents§3The power-screw equations
A power screw is the inclined plane with friction, solved: one expression for the torque to raise the load, its mirror for lowering, and the lead and friction angles doing all the work.
The reading is geometric. Raising the load is pushing it up the helix’s incline against friction, so the effective slope the torque must climb is the lead angle plus the friction angle. Lowering reverses the sign: gravity now helps, friction still resists, and the torque required is proportional to tan(ρ′ − λ) — a quantity whose sign is the self-locking test of the systems page. Positive, and the screw must be actively driven downward: it holds any load unpowered. Negative, and the load overhauls the screw — it back-drives, and something else must hold it. The flank angle enters only through the small correction μ′ = μ/cos φ, the same vee-deepening factor as before, and for the trapezoid’s 15° it costs almost nothing: cos 15° = 0.966, a 3.5% friction surcharge against the vee thread’s 15.5%. Everything a jack, a vice, a press or a machine slide does is these two lines evaluated — which is precisely what §4 now does, on the hero’s thread.
Contents§4A Tr30×6 jack, worked
Put 10 kN on a Tr30×6 screw with an oiled bronze nut and every number a jack designer needs falls out of §3 in six lines.
| Quantity | Formula | Value |
|---|---|---|
| Pitch diameter | d2 = d − p/2 | 27 mm |
| Lead angle | λ = atan(p / π d2) | 4.046° |
| Friction angle | ρ′ = atan(0.12 / cos 15°) | 7.08° |
| Torque to raise | F (d2/2) tan(λ + ρ′) | 26.6 N·m |
| Torque to lower | F (d2/2) tan(ρ′ − λ) | 7.2 N·m |
| Efficiency, raising | tan λ / tan(λ + ρ′) | 36.0% |
| Read the two torques together. Raising ten kilonewtons — a tonne — takes 26.6 N·m, comfortable spanner effort: the inclined plane at work. Lowering takes +7.2 N·m of driven torque, and that positive sign is the whole safety case: with λ at 4.05° against ρ′ at 7.08°, the screw is self-locking, the tonne sits on the thread indefinitely with the handle removed, and a jack needs no brake, pawl or lock — its own inefficiency is the brake. (A real jack adds collar friction under the load cap, raising both torques; the thread-only figures are the mechanism laid bare.) | ||
§5Living with 36%
The 64% that never becomes lift does not vanish — it becomes heat and wear at the nut flanks, and designing a power screw is mostly designing for that fact.
At §4’s efficiency, every unit of lifting work is bought with about 1.78 units of friction heat (1/0.360 − 1), generated in the thin oil film and the bearing flanks of the nut. That single ratio explains the standing conventions of lead-screw practice. The nut is bronze on a steel screw — the dissimilar, embeddable, low-friction pairing the plain-bearings page of the next section will formalise — because the nut is a bearing that happens to be helical, and it is made the sacrificial partner: screws are long and expensive, nuts are short and replaceable. Lubrication is not optional maintenance but part of the machine’s rating, and duty cycle matters: a screw that positions occasionally can run hard, while one that strokes continuously must be sized for heat, not strength. And wear arrives as backlash — lost motion at every reversal as the load crosses the worn gap between flanks — which the trapezoid’s split-nut adjustment of §2 exists to recover, and which sets the service life of every hand-adjusted machine slide. None of this is failure; it is the agreed price of §4’s self-locking. The screw that cannot run away is the screw that runs warm.
Contents§6Acme, Tr and the ball screw
The 29° Acme and 30° Tr are twins that never intermarry — and both now share the field with the ball screw, which repealed the friction tax and lost the free brake in the same stroke.
The forms differ by one degree of included angle and their pitch tables, which is close enough to tempt and far enough to ruin: an Acme nut on a Tr screw bears on mismatched flanks and pitches, and the pairing rule is absolute, as everywhere in this section. Between them they own the classical territory — jacks, vices, presses, clamps, valve stems, machine-tool slides — everywhere §4’s self-locking is worth §5’s heat. The ball screw changes the bargain entirely: recirculating balls replace sliding flanks with rolling contact, efficiency climbs to around 90%, and the heat and wear economy of §5 largely disappears — which is why every CNC axis rides one. But run §3 backwards at that efficiency and the self-locking is gone with the friction: a loaded ball screw back-drives freely, a vertical axis falls when the drive releases, and the brake the Acme carried inside its own geometry must now be bolted on as a component. That is the cleanest summary this page can offer of the whole power-thread trade — friction is a tax and a brake in the same coin — and the ball-and-Acme-leadscrews page of the machine-elements section takes the story on from here.
Contents§7Quick reference
The working core of the page on one card rack.
Form
Acme 29° · Tr 30°
depth p/2 · flats ≈ 0.37 p
Equations
T = F(d2/2) tan(λ ± ρ′)
μ′ = μ / cos 15°
Tr30×6 · 10 kN
raise 26.6 · lower 7.2 N·m
η = 36.0%, self-locking
The price
≈1.78 heat per unit lift
bronze nut · oil · backlash
Ball screw
η ≈ 90%, back-drives
the brake moves outside