Power Screw (Square Thread)

torque-power

Verified build 4 relations · 4 identities proven · 1 modeling step · 6 parity samples

Car jacks, bench vises, lathe lead screws, linear actuators, clamping fixtures, piano stools — whenever a machine needs to turn modest torque into crushing linear force and hold it there with the power off, it reaches for a power screw. The whole device is one idea: an inclined plane wrapped around a cylinder. Unroll one turn of thread and you are pushing a block up a ramp of rise $l$ and run $\pi d_m$ :

T_R = \frac{F d_m}{2}\!\left(\frac{l + \pi f d_m}{\pi d_m - f l}\right) \qquad T_L = \frac{F d_m}{2}\!\left(\frac{\pi f d_m - l}{\pi d_m + f l}\right) \qquad e = \frac{F\,l}{2\pi T_R}

Three things to notice, one per formula:

Friction is a knob here, not a nuisance. Unlike every elastic THING in this catalog, the governing input is a surface property — and an honest model treats it as the uncertain dial it is (dry steel-on-steel, greased, bronze nut: roughly 0.06 to 0.25, and it changes as the grease ages). Drag $f$ through that range in the widget and watch the torque double. This is why real actuators are sized with friction margins, not friction estimates.
Self-locking is a sign change you can read. While $\pi f d_m > l$ , the lowering torque $T_L$ is positive: the load stays put and you must actively screw it down — a jack. The moment the lead outgrows friction, $T_L$ flips negative and the load will happily spin the screw on its own — a back-drivable lead screw (this is how some “self-feeding” drills and yo-yo-like ball screws behave on purpose). One subtraction in the numerator, two completely different machines.
Locking is purchased with efficiency. At the self-locking boundary $f = \tan\lambda$ the efficiency is already below 50 %, and a comfortably locking jack runs ~30–40 %: most of your cranking becomes heat in the nut. Ball screws exist precisely to refuse this bargain — ~90 % efficient, and they need a brake for exactly the reason a jack doesn’t.

The capacity configuration runs the relations backwards: give it your motor’s torque budget and it reports the load the screw can raise — then $T_L$ tells you whether that load stays raised.

Try it

Governing relations

\tan\lambda = \frac{l}{\pi d_m}

Assumes: unwrap one turn of thread at the mean diameter — the thread IS an inclined plane of rise l and run πd_m; single-start thread (for multi-start screws, l is the lead, not the pitch)

Source: Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §8-2 "The Mechanics of Power Screws": Fig. 8-6 (unrolled-thread free bodies), eq. (8-1) torque to raise, eq. (8-2) torque to lower, eq. (8-3) self-locking condition πf d_m > l ⇔ f > tan λ, eq. (8-4) efficiency e = Fl/(2πT_R).

T_R = \frac{F d_m}{2}\left(\frac{l + \pi f d_m}{\pi d_m - f l}\right)

Assumes: square thread — the normal force is not tilted by a thread half-angle (Acme threads divide the friction terms by cos 14.5°); thrust-collar friction neglected (rolling-element collar) — a plain collar can add more torque than the thread itself; steady, slow raising — friction fully developed, no dynamics · Valid while: The wedge jams: f·tan λ ≥ 1, so the harder you twist, the harder the thread presses itself into the nut — no finite torque raises the load. A thread this steep and grippy is not a power screw; back off the lead or the friction.

T_L = \frac{F d_m}{2}\left(\frac{\pi f d_m - l}{\pi d_m + f l}\right)

Assumes: same square-thread, no-collar model with the load now helping the motion — positive T_L is torque you must apply to let the load DOWN under control · Valid while: Not self-locking — T_L has gone negative: the load can spin the screw and lower itself the moment the drive lets go (f < tan λ). Fine for a lead screw you back-drive on purpose; alarming for a jack. Hold it with a brake or pick a finer lead.

e = \frac{F\,l}{2\pi\,T_R}

Assumes: useful work per turn (F·l) over input work per turn (2π·T_R); the frictionless screw (f = 0) would need T_0 = F·l/2π, so e = T_0/T_R

Derivation

Steps marked modeling step are where physics enters by citation — every other line is machine-proven to follow from them, and the cited models are independently re-derived in the test pipeline where possible. See what is and isn't machine-verified.

\tan{\left(lam \right)} = \frac{l}{\pi d_{m}}

1. The modeling step: cut the thread along the mean diameter and unroll one full turn onto the page. What was a helix becomes an inclined plane — base πd_m (one circumference), rise l (one lead). Raising the load is pushing a block up this plane; the lead angle λ is the plane's slope. Everything below is freshman statics on that block. — unwrap the helix: thread = inclined plane modeling step

P_{R} \left(- f \sin{\left(lam \right)} + \cos{\left(lam \right)}\right) = F \left(f \cos{\left(lam \right)} + \sin{\left(lam \right)}\right)

2. Equilibrium of the block sliding up the plane: a horizontal push P_R, the weight F, and a fully-developed friction force f·N pointing down-slope. Summing forces along and across the plane and eliminating N gives P_R = F(sin λ + f cos λ)/(cos λ − f sin λ). (The build verifies this line is exactly equivalent to the closed form authored for T_R; the test pipeline re-derives it from the two equilibrium equations with N kept explicit.) — statics: block on the plane, eliminate N

T_{R} = \frac{P_{R} d_{m}}{2}

3. The horizontal push acts at the thread's mean radius, so the torque is P_R·d_m/2. Dividing the equilibrium result through by cos λ and substituting tan λ = l/(πd_m) turns the trigonometry into hardware numbers: T_R = (F d_m/2)·(l + πf d_m)/(πd_m − f l). — torque = push × mean radius

T_{L} \left(\pi d_{m} + f l\right) = \frac{F d_{m} \left(\pi d_{m} f - l\right)}{2}

4. Run the same block DOWN the plane: gravity now helps and friction switches sides. The numerator πf d_m − l is the whole self-locking story in one subtraction — while friction outweighs the lead, T_L is positive (you must push to lower the load); the moment the lead wins, T_L flips sign and the load drives the screw. Set f to zero in the widget and watch a jack become a fan. — statics: same block, descending

2 \pi T_{R} eff = F l

5. One turn of the screw does 2πT_R of work at the handle and lifts the load by exactly one lead, F·l of useful work. The ratio is the efficiency — and because the frictionless torque is T_0 = Fl/2π, this is also T_0/T_R. Square-thread jacks routinely run 30–50 %: the rest of your effort is heating the nut. A self-locking screw is provably under 50 % — locking is purchased with friction, and friction is purchased with work. — work out / work in

How it fails

The widget models torque equilibrium on a perfect square thread. Real power screws fail around the edges of that model:

Nut wear is the design driver. The thread is a bearing that slides under full load every stroke; the classic pairing (steel screw, bronze nut) sacrifices the cheap nut deliberately. Wear is governed by bearing pressure $F/(\pi d_m h \cdot n_t)$ and sliding speed — the $pV$ limit — which is why jack threads are fat and slow. A worn nut doesn’t just rattle: lost flank contact concentrates load on fewer threads and accelerates the end.
The first thread carries the load. In any screw–nut pair the load does not share equally; elastic mismatch piles a third or more of the total onto the first engaged thread. Thread stripping starts there, and adding more engagement length past ~3 threads buys surprisingly little.
Heat. A 35 %-efficient screw turns two-thirds of the input work into heat at the thread interface. Long strokes or rapid cycling cook the grease out, friction climbs, torque climbs, galling begins — a feedback loop that seizes screws mid-stroke. Acme jacks are rated by duty cycle for this reason.
Self-locking is not a brake. The static condition $\pi f d_m > l$ holds only while the friction is the static, greasy value you assumed. Vibration is the classic thief: each shake momentarily unloads the thread, the effective $f$ falls toward its (smaller) kinetic value, and the load ratchets down. A vibrating self-locking jack creeps. Anything carrying people gets a positive mechanical lock, not a friction promise.
The screw is also a column. A long screw raising a load is a slender column under compression (Euler’s problem) — jack screws are short and fat as much for buckling as for wear. And the body of the screw still carries torsion plus direct compression: the combined stress state, not the thread torque, sizes the core diameter.
Backlash and elasticity. Thread clearance plus wind-up of a long screw under torque makes positioning hysteresis — a lead screw under reversing load takes up its backlash with a clunk. Machine tools either preload split nuts or switch to ball screws and accept needing a brake.

Belt Drive (Flat Belt / Capstan)
Friction compounding like interest: every degree of wrap multiplies the tension a belt can hold, e^μθ in total — the same exponential that lets a sailor check a ship with two turns of rope. At speed, centrifugal relief steals tension back, so every belt has a power ceiling.
- torque-power
Planetary (Epicyclic) Gearset
Three coaxial members — sun, ring, planet carrier — share one gear mesh law. With two degrees of freedom, it has no single "ratio": fix a different member and the same hardware becomes a different transmission.
- kinematics
- torque-power
Spur Gear Pair (Lewis Bending)
Two meshing spur gears carry power through one tangential tooth force. The Lewis equation turns that force into a root-bending stress, and a cited form-factor table sets how much a tooth of N teeth can take. Same load, same module — yet the pinion, with fewer teeth, always works harder.
- stress
- torque-power
Axial Disk Clutch / Brake (Uniform Wear vs Uniform Pressure)
The torque an axial plate clutch can pass depends on an assumption you cannot see: how the contact pressure is distributed across the friction annulus. A new, rigid clutch presses uniformly; a worn-in one wears until pressure ∝ 1/r, concentrating load at the inner edge. This page shows both torque predictions side by side — never picking a winner — with the worn-in model always giving the smaller (safe) number, and the r_i = r_o/√3 that squeezes the most torque from a given lining.
- torque-power
Bolted Joint with Gasket (External Tensile Load)
A preloaded bolt clamping a gasketed joint, then pulled by an external tensile load. The bolt and the members act as two springs in parallel, so the external load does NOT all go to the bolt — it splits by stiffness. The build solves the coupled bolt/member force system exactly and refuses the moment the members go slack.
- stress
Helical Compression Spring
A torsion bar wound into a package: push on the coil and the wire twists. G sets the rate, σ_y sets the margin, and the geometry trades them against three envelopes — coil bind, buckling, and a spring index you can actually wind.
- stiffness
- stress
- mass-cost

Chains with

Outputs whose SI dimension and quantity kind match another THING's input — the only wires the planner's connectionLegal accepts (invariant 2, computed at build time, not hand-listed). Wire these on the chaining demo.

Composite Bar (Core + Sleeve)
- F P
Symmetric Two-Bar Truss
- F P
- lam alpha
Cantilever Beam (End Load)
- F P
Curved Beam in Bending (Winkler — Crane Hook, C-Clamp, Press Frame)
- F P
Simply Supported Beam (Center Load + UDL)
- F P
Transverse Shear in Beams (τ = VQ/Ib, Shear Flow, Fastener Spacing)
- F V
Fixed-Fixed Torsion Shaft (Interior Torque)
- T_L T
- T_R T
Rectangular Shaft in Torsion (Saint-Venant)
- T_L T
- T_R T

+ 17 more THINGs its outputs can legally feed (showing the first 8 in course order).

Sources

Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §8-2 "The Mechanics of Power Screws": Fig. 8-6 (unrolled-thread free bodies), eq. (8-1) torque to raise, eq. (8-2) torque to lower, eq. (8-3) self-locking condition πf d_m > l ⇔ f > tan λ, eq. (8-4) efficiency e = Fl/(2πT_R).