Power Screws and Lead Screws
Thread geometry, torque to raise and lower a load, efficiency, self-locking and ball screws.
What a power screw does
A power screw turns rotation into straight-line motion and a small torque into a large axial force: jacks, presses, vices, valve actuators, machine tool slides and linear stages. The mechanical advantage is large (the force is 2π ÷ lead times the torque, less friction) and the motion slow and smooth. The cost is efficiency: a sliding screw is only 25 to 45 percent efficient, a ball screw 90 percent. The screw’s counterpart in Motion Control and Actuators is the ball screw axis (see the sizing page there); this page gives the machine-element view of the thread, friction and limits.
Thread geometry
The pitch p is the distance between adjacent threads; the lead l is the distance the nut moves in one revolution (l = p for a single-start thread, 2p for a double start). The lead angle is λ = arctan(l ÷ (π dm)), with dm the mean (pitch) diameter. A trapezoidal Tr 30 × 6 screw has a major diameter of 30 mm, a pitch of 6 mm, a mean diameter dm = 30 − 6 ÷ 2 = 27 mm, a lead angle of λ = arctan(6 ÷ (π × 27)) = 4.05° and a root (minor) diameter of 23 to 24 mm.
Torque to raise and lower a load
Traise = (W dm ÷ 2) × (l + π μ dm) ÷ (π dm − μ l) Tlower = (W dm ÷ 2) × (π μ dm − l) ÷ (π dm + μ l)
Worked example: the press stop screw
Worked example: a Tr 30 × 6 screw raising 10 kN against the press stop
- Mean diameter 27 mm, lead 6 mm: λ = 4.05° (tan λ = 0.0707).
- With μ = 0.15 (steel on bronze, lubricated): Traise = 10 kN × 13.5 mm × (6 + π × 0.15 × 27) ÷ (π × 27 − 0.15 × 6) = 30.1 N·m at the thread.
- Efficiency of the thread e = W l ÷ (2π T) = 10 000 × 6 ÷ (2π × 30 119) = 32 percent; the other 68 percent is friction heat.
- Collar friction (a thrust washer with μc = 0.10 at a mean radius of 20 mm): Tc = 0.10 × 10 000 × 20 = 20.0 N·m. Total torque 50.1 N·m, overall efficiency 19 percent. A ball thrust bearing would cut the collar term to under 1 N·m.
- Speed and power: at 300 rpm the nut moves 30 mm/s; the input power is T ω = 50.1 × 31.4 = 1 575 W, while the useful power is W v = 300 W: the difference is heat in the nut (check the nut’s pressure-velocity limit with the maker).
- Lowering: Tlower = 10.6 N·m is positive, so the load does not drive the screw back down: the screw is self-locking.
Efficiency and self-locking
The screw self-locks (holds the load with no brake) when the friction angle exceeds the lead angle, i.e. μ > tan λ. Here μ = 0.15 > 0.071: it locks. The consequence is a trade-off: a screw that holds its load has an efficiency below 50 percent. An efficient screw (ball screw, or a steep lead) will back-drive: the load turns it, so the drive needs a brake or a worm stage. Vibration can unlock a marginally self-locking screw; for hoisting use a brake or a second locking element.
Ball screws
A ball screw replaces sliding with rolling: recirculating steel balls between the nut and screw. Friction coefficient falls to 0.005 to 0.02, efficiency to about 90 percent, and wear and backlash are low (preloaded nuts remove the backlash). The torque for the same load: T = W l ÷ (2π e).
Worked example: a ball screw in place of the Acme screw
- The 6 mm lead ball screw at e = 0.90 needs T = 10 000 × 6 ÷ (2π × 0.90) = 10.6 N·m, 0.2 times less than the 50.1 N·m of the Acme screw with its collar (4.7× the torque), so a much smaller motor and no heating problem.
- A 10 mm lead ball screw needs 17.7 N·m for the same load but moves 10 mm per turn: 1.7 times the speed per rpm.
- Back-driving: the load drives the 10 mm lead screw back with a torque of W l e ÷ (2π) = 14.3 N·m: a vertical axis needs a brake (a spring-applied holding brake on the motor) or it falls when the power fails.
Buckling, critical speed and thread stress
- Column buckling: a long screw in compression buckles like a column: Pcr = π² E I ÷ L² for pinned ends, with I from the root diameter. For dr = 24 mm and L = 500 mm: I = 16 286 mm⁴, Pcr = 135 kN (540 kN with fixed ends): against 10 kN a factor of 13.5 for the pinned case; use a safety factor of at least 3 to 5 (see Columns and Buckling).
- Critical speed: a long, rotating screw whips at its natural frequency. For a 26 mm root diameter screw 1 m between pinned supports, ω = (π ÷ L)² × (dr ÷ 4) √(E ÷ ρ) = 332 rad/s = 3 169 rpm; run below 80 percent of that: 2 535 rpm, i.e. 25.3 m/min with a 10 mm lead. Fixed ends raise the limit by 1.5 to 2.2 times (see Vibration, Isolation and Noise).
- Thread stress: axial stress in the root area 661 mm² is 15 MPa (plus torsion); the bearing pressure on the nut thread: W ÷ (π dm × thread depth × number of threads) ≈ 6.5 MPa for a 36 mm nut, below the 5 to 10 MPa limit for bronze (3 to 5 for a polymer nut).
- Wear and lubrication: use a lubricant film and a nut of bronze or filled plastic that wears before the screw; backlash grows with wear (a split nut or a spring-loaded one takes it up).
Choosing a screw
| Type | Friction μ | Efficiency | Notes |
|---|---|---|---|
| Square | 0.10 to 0.20 | 35 to 50 % | rarely made now; high strength; difficult to adjust for wear |
| Acme / trapezoidal (Tr) | 0.10 to 0.20 | 25 to 45 % | the common lead screw; split nuts take up wear; self-locking at low leads |
| Buttress | 0.10 to 0.20 | 30 to 50 % | one-way heavy thrust (presses, vices) |
| Ball screw | 0.005 to 0.02 | 85 to 95 % | recirculating balls; precise; not self-locking; needs a brake |
| Roller screw | 0.003 to 0.01 | 85 to 95 % | highest load, speed and life; expensive |
Related
- Previous: Seals, O-Rings and Gaskets. Next: Cams and Followers.
- Mechanical Transmission (ball screws) and Sizing a Motion Axis.
- Bolted and Welded Joints for fastener threads.
- Machine Design: the whole guide.
FAQ
What is the difference between pitch and lead?
Pitch is the spacing of the thread; lead is the travel per revolution. They are equal for a single-start thread; a two-start thread has a lead of twice the pitch.
Can I back-drive a lead screw?
Not if it self-locks (μ > tan λ): that is why jacks hold. A ball screw or a steep-lead screw will back-drive and needs a brake.
Why does my screw get hot?
Friction: a sliding screw dissipates 55 to 75 percent of the input power. Reduce the speed, lubricate, use a better nut material or a ball screw.
Why does the axis wobble at speed?
You are near the critical speed of the screw. Shorten the span, increase the diameter, support the far end or reduce the speed.
References
- Shigley, J. E., Mischke, C. R., Budynas, R. G., Mechanical Engineering Design, McGraw-Hill (power screws); Norton, R. L., Machine Design, Pearson.
- ISO 2904 (trapezoidal thread profile), ISO 3408 (ball screws), DIN 103 (Tr threads); THK, NSK, Hiwin and Bosch Rexroth ball screw catalogs.
- Machinery’s Handbook, Industrial Press (screw threads, Acme and trapezoidal).
Last reviewed: October 2026. Friction coefficients and efficiencies are typical for lubricated screws.