A Screw Pair is a lower kinematic pair where one element — a threaded shaft — rotates inside a matching threaded nut, and that rotation forces a linked translation along the shaft axis. The motion is helical: every revolution advances the nut by exactly one lead. The purpose is to convert rotary input into precise, high-force linear output with strong mechanical advantage. You see this in machine-tool lead screws, vises, jacks, and Linear Actuator drives where 1 turn can move a 1,000 lb load by 5 mm.
Screw Pair Interactive Calculator
Vary screw pitch, thread starts, rpm, and run time to see lead, linear speed, revolutions, and travel update with an animated screw-pair diagram.
Equation Used
The screw pair converts rotation to linear motion through the screw lead. Lead is pitch times the number of starts. Linear speed is lead times rotational speed, with rpm divided by 60 to convert revolutions per minute to revolutions per second.
- Lead equals pitch times number of thread starts.
- Screw rpm is steady and the nut is constrained against rotation.
- No allowance is included for backlash, slip, compliance, or acceleration.
Inside the Screw Pair
The Screw Pair pins two bodies together with a single helical constraint. Spin the shaft and the nut walks along it — that coupled rotation-and-translation is what Reuleaux classified as a lower pair, because the contact is surface-on-surface across the entire engaged thread length, not point or line contact. One degree of freedom: either rotation or translation, and the helix angle locks the ratio between them. That ratio is the lead, and it sets everything — speed, force, and whether the screw will back-drive under load.
Screw Threads do the actual work. A standard Acme thread runs a 29° flank angle and gives you good load capacity with reasonable friction. A ball screw replaces sliding contact with recirculating bearings and pushes efficiency from around 30% on Acme up to 90%+. The Screw Movement is dictated entirely by pitch (distance between adjacent thread crests) and the number of starts. A single-start screw with 5 mm pitch advances 5 mm per turn. A 4-start screw at the same 5 mm pitch advances 20 mm per turn — same physical thread, four times the lead.
Get the tolerances wrong and the pair fails in predictable ways. Backlash above 0.1 mm on a CNC lead screw shows up as positioning error every time the axis reverses direction. Run a sliding screw dry and friction climbs from 0.10 to 0.25 within hours, the nut overheats, and the bronze galls onto the steel shaft. Side-load the nut beyond about 5% of rated axial load and the threads contact unevenly — you get accelerated wear on one flank and the screw starts to whip at higher RPM.
Key Components
- Threaded Shaft (Screw): The rotating element carrying the helical thread. Material is typically hardened C45 steel or 4140 for power screws, or precision-ground bearing steel for ball screws. Straightness must hold to 0.05 mm/m to avoid whip above 1,500 RPM.
- Nut: The translating element with matching internal thread. Bronze (C932) for sliding Acme nuts, hardened steel with recirculating balls for ball screws. Fit class 2G or 3G on Acme threads gives 0.05–0.15 mm radial clearance — tighter binds, looser backlashes.
- Thread Form: The cross-section profile — Acme (29°), trapezoidal metric (30°), buttress (asymmetric, for one-direction loads), or square (highest efficiency, hardest to manufacture). Form choice sets friction coefficient and load rating.
- Lead: Axial distance per revolution. Equals pitch × number of starts. Sets the speed-to-force tradeoff: short lead means high force and self-locking, long lead means high speed and back-drivable.
- End Bearings: Support the shaft against thrust and radial loads. Angular contact bearings preloaded to roughly 8% of dynamic capacity eliminate axial play. Without proper preload you'll see 0.02–0.05 mm of axial slop reflected directly in your linear position.
Who Uses the Screw Pair
The Screw Pair shows up anywhere you need to convert a motor's rotary output into controlled linear force. The Screw is one of the oldest mechanisms in continuous use — Archimedes wrote about it, and it still drives the Z-axis on a $400,000 Mazak machining centre. Different industries lean on different variants: machine tools want precision, presses want force, and consumer electronics want compactness.
- Machine Tools: CNC milling machines like the Haas VF-2 use ground ball screws on X, Y, and Z axes — 16 mm diameter, 5 mm lead, holding ±0.005 mm positioning accuracy over a 762 mm stroke.
- Linear Motion: FIRGELLI Linear Actuator products use Acme lead screws driven by DC gearmotors — a 12 V actuator with a 4 mm-pitch screw and 50:1 gearbox produces 200 lb of axial force at 0.5 in/s.
- Aerospace: Boeing 737 horizontal stabilizer trim is driven by a jackscrew assembly — a 38 mm Acme screw with a primary and backup nut, infamously implicated in the Alaska Airlines 261 accident when wear exceeded the 0.040 in service limit.
- Manufacturing Presses: Bench-top arbour presses and screw-type C-clamps multiply hand input to several thousand pounds of clamping force using single-start 1/2-13 UNC threads.
- Medical Equipment: Hospital bed height adjustment and patient-lift hoists use self-locking trapezoidal screws — the lead angle stays below the friction angle so the load holds position without a brake when power drops.
- 3D Printing: Prusa i3 and similar FDM printers use 8 mm T8 lead screws with 8 mm lead on the Z-axis, giving 200 microsteps per revolution = 0.04 mm per microstep at typical 1/16 microstepping.
The Formula Behind the Screw Pair
The core relationship for a Screw Pair connects rotation to linear travel through the lead. This single equation tells you how fast the nut moves and, when combined with the friction angle, whether the screw will hold load when the motor stops. At the low end of typical operating range — say a 2 mm-lead jackscrew at 30 RPM — the nut creeps at 1 mm/s, slow enough that you can watch the load inch up. At the high end — a 20 mm-lead ball screw at 3,000 RPM — you're pushing 1,000 mm/s and now critical-speed whip becomes the limit, not the math. The sweet spot for general-purpose power screws sits around 4–10 mm lead at 500–1,500 RPM, where you get usable speed without entering whip territory or overheating the nut.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| v | Linear velocity of the nut | mm/s | in/s |
| L | Lead — axial distance per revolution | mm/rev | in/rev |
| N | Rotational speed of the screw | RPM | RPM |
| η | Mechanical efficiency (Acme ≈ 0.30, ball screw ≈ 0.90) | dimensionless | dimensionless |
Worked Example: Screw Pair in a CNC router Z-axis lead screw
You're sizing a Z-axis on a hobby CNC router. The plan is a 12 mm Acme lead screw with 4 mm lead, single-start, driven by a NEMA 23 stepper at a nominal 600 RPM. You need to know the linear feed rate, what happens when you slow it for fine plunges, and what happens if you push it to top speed for rapid moves.
Given
- L = 4 mm/rev
- Nnom = 600 RPM
- Nlow = 120 RPM
- Nhigh = 1,200 RPM
- η = 0.32 Acme sliding
Solution
Step 1 — compute the nominal linear velocity at 600 RPM:
That's 2,400 mm/min, a normal cutting feed rate for plunging into MDF or soft aluminium with a 6 mm end mill. The motor sees comfortable torque demand here and the nut runs cool.
Step 2 — at the low end, 120 RPM for a precision finishing plunge:
This is 480 mm/min — slow enough to see the chip form. You'd use this for the final 0.2 mm depth pass when surface finish matters. Below about 60 RPM the stepper starts losing low-speed smoothness and you may see vertical banding in the cut.
Step 3 — at the high end, 1,200 RPM for rapid traverse between cuts:
That's 4,800 mm/min in theory. In practice, a 12 mm Acme screw running unsupported at 1,200 RPM is approaching its critical whip speed for spans over about 400 mm, and an unlubed sliding nut at this speed will heat 30–40°C above ambient within a minute of continuous run. Most builders cap rapids on this setup at around 900 RPM (60 mm/s) for thermal headroom.
Result
Nominal Z-axis feed rate is 40 mm/s (2,400 mm/min) at 600 RPM. That's a workable cutting speed where the motor isn't straining and the nut isn't cooking. Across the range, 8 mm/s gives you fine-finish control while 80 mm/s is the theoretical ceiling — but the practical sweet spot is 40–60 mm/s, because beyond that whip and nut friction degrade accuracy faster than the speed gain helps. If your machine measures 30 mm/s instead of the predicted 40 mm/s, check three things: stepper missing steps under acceleration load (listen for the characteristic clack), coupling slip between motor shaft and screw end (a loose set screw on a flat is the classic offender), or excessive nut preload causing friction torque the motor can't sustain at speed.
Choosing the Screw Pair: Pros and Cons
The Screw Pair competes against rack-and-pinion and belt-driven linear motion for any rotary-to-linear job. Each wins on different axes, and the right choice depends on whether you prioritise force, speed, cost, or precision. Here's the practical comparison.
| Property | Acme Lead Screw | Ball Screw | Rack and Pinion |
|---|---|---|---|
| Efficiency | 25–40% | 85–95% | 80–90% |
| Typical max speed | 50 mm/s | 1,000 mm/s | 5,000 mm/s |
| Positioning accuracy (per metre) | ±0.10 mm | ±0.01 mm | ±0.05 mm |
| Self-locking under load | Yes (lead angle < friction angle) | No (back-drives freely) | No |
| Load capacity | Up to 50 kN | Up to 200 kN | Up to 500 kN |
| Relative cost (1 m assembly) | $50–150 | $400–1,500 | $200–800 |
| Maintenance interval | Re-grease every 100 hours | Re-grease every 500–2,000 hours | Open lubrication, weekly |
| Best application fit | Low-speed power, vises, jacks | CNC, precision linear stages | Long-stroke, high-speed gantries |
Frequently Asked Questions About Screw Pair
Almost always misalignment between the screw axis and the linear guide axis. On the bench the nut floats freely; once you bolt the nut block to a carriage running on rails, any parallelism error above about 0.05 mm over the screw length forces the nut to fight the rails. The nut tries to follow the screw, the carriage tries to follow the rails, and the friction spike happens wherever the misalignment peaks.
Quick diagnostic: loosen the nut-block mounting bolts and run the screw end to end. If it's smooth, your nut block needs a floating mount — a single-point flexible coupling or a slotted mounting plate that lets the nut self-centre on the screw.
Yes — Screw Pair is the kinematic classification (Reuleaux's lower pair), and lead screw, power screw, jackscrew, and ball screw are all real-world implementations of that same pair. The kinematics are identical: one helical degree of freedom coupling rotation to translation. What changes between them is the thread form, the contact mechanism (sliding versus rolling balls), and the efficiency. When engineers say "lead screw" they usually mean a sliding Acme or trapezoidal screw used for motion; "power screw" emphasises force transmission; "ball screw" is the rolling-element version.
Compare the lead angle to the friction angle. Lead angle α = arctan(L / (π × d)), where L is lead and d is the pitch diameter. Friction angle φ = arctan(μ), where μ is the coefficient of friction (around 0.10–0.15 for steel-on-bronze, lubricated). If α < φ, the screw is self-locking and will hold load without a brake. If α > φ, it back-drives and you need a brake or a worm gear upstream.
Rule of thumb: most single-start Acme and trapezoidal screws under 12 mm pitch are self-locking. Ball screws are essentially never self-locking — efficiency is too high.
Acme, almost certainly. The application is vertical, low duty cycle, and the load wants to fall when power drops. Self-locking is a feature here — a ball screw would back-drive and drop the load the moment the motor de-energises, forcing you to add a brake (cost, weight, failure mode). An Acme screw with a 4 mm lead at 600 RPM gives 40 mm/s lift speed and holds position dead with no power.
Switch to ball screws when speed matters more than holding (CNC axes), when efficiency drives motor sizing (battery-powered systems), or when you're cycling thousands of times per hour and Acme nut wear becomes the bottleneck.
You've hit the critical speed of the unsupported shaft. Every screw has a natural bending frequency that scales with diameter squared and inversely with span squared. A 12 mm screw on 500 mm centres typically whips around 1,500–1,800 RPM in fixed-simple support. Above that, the shaft bows out under its own centripetal force and you see violent lateral vibration.
Fixes in order of effectiveness: shorten the unsupported span, increase shaft diameter (a 16 mm screw raises the limit by roughly 1.8×), change end-fixity from simple-simple to fixed-fixed (raises critical speed by 2.2×), or operate below 80% of the calculated critical speed as a safety margin.
For a bronze Acme nut running at typical loads (under 30% of rated dynamic capacity), every 100 operating hours of continuous use, or every 500 cycles for intermittent duty. Past that interval, the EP grease film breaks down, sliding contact goes boundary-lubricated, and bronze starts shedding into the thread valleys.
The wear signature is asymmetric: backlash grows fastest on the loaded flank. If you measure 0.05 mm new-build backlash and it's at 0.20 mm after a few hundred hours, the nut is roughly half worn out. Anti-backlash spring-loaded nuts buy you another 30–50% service life by taking up wear automatically — worth the extra cost on any positioning axis.
You probably used the ideal efficiency in the calculation. Real Acme screws under modest preload run 25–35% efficient, not the 40–50% you find in textbook charts. Plug η = 0.30 into F = (2π × T × η) / L and your predicted force drops to match what you're measuring.
Other contributors: thrust bearing friction (subtract another 5–10% if the bearing is unlubricated or oversized), nut preload eating into the available output torque, and motor torque rolloff at speed — a stepper at 600 RPM puts out 40–60% of its holding torque. Always size with η = 0.25 for sliding screws to leave headroom.
References & Further Reading
- Wikipedia contributors. Leadscrew. Wikipedia
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