Inputs
Pitch diameter d₂, pitch P and number of starts n, the axial force F, the nut material with its lubrication state (or a custom friction angle), and optionally the friction diameter and coefficient of the thrust face and the speed.
Threads & bolts · Power screws
Power screw: torque, efficiency and self-locking
Calculate the torque a power screw needs to raise and lower its load, its efficiency, and the decisive question of whether it is self-locking. The friction angle comes from a reference table by nut material and lubrication and can be replaced by your own value at any time. Feed rate and drive power are added on request.
Torque, efficiency and self-locking are calculated. Thread flank pressure, buckling safety of the spindle, the combined compressive and torsional stress and the wear life of the nut have to be verified separately.
ηφ vs. ρ′
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Torque, efficiency and self-locking
Set the screw and the load.
Inputs and method
Power screw to Roloff/Matek
Calculate the drive torque, efficiency and self-locking of a lead screw, and the torque needed to lower the load.
Calculation
From tan φ = Ph/(π·d₂) with Ph = P·n follows the lead angle. The required torque is T = F·[d₂/2·tan(φ ± ρ′) + d_L/2·μ_L], plus for raising and minus for lowering; the thrust friction resists motion both ways and is therefore always added. The efficiency is η = tan φ/[tan(φ + ρ′) + μ_L·d_L/d₂]. The screw is self-locking while φ < ρ′.
Example
A Tr 20×4 screw (d₂ = 18 mm) at 10 kN axial force with a lubricated bronze nut (ρ′ = 6°) has φ = 4.05° and is therefore self-locking. Raising requires 15.94 N·m at 39.9 % efficiency; lowering requires 3.07 N·m to be applied, because the load does not come down by itself.
Sources and limits: Roloff/Matek, Maschinenelemente Formelsammlung, 15th edition, section 8 "Bewegungsschrauben" (no. 72 torque, no. 81 efficiency and self-locking); reference values for ρ′ by nut material and lubrication from the same source. Only torque, efficiency and self-locking are calculated – flank pressure, buckling, combined stress and wear must be verified separately.
Power screw to Roloff/Matek in detail
Calculate the drive torque, efficiency and self-locking of a lead screw, and the torque needed to lower the load.
What is a power screw?
A power screw converts rotation into linear motion and, with it, torque into a large axial force. Unlike a fastener, which is tightened once, it runs under load – so the questions are not about preload but about torque, efficiency and whether the load stays put once the drive stops. The answer to that lies entirely in two angles: the lead angle of the thread and the friction angle of the pairing.
Formula and variables
T = F · [d₂/2 · tan(φ ± ρ′) + d_L/2 · μ_L]
tan φ = Ph / (π · d₂) with Ph = P · nRaising: +ρ′ · Lowering: −ρ′ · Thrust friction always additiveη = tan φ / [tan(φ + ρ′) + μ_L · d_L/d₂]Self-locking for φ < ρ′
| Symbol / input | Meaning |
|---|---|
| d₂, P, n | Pitch diameter, pitch and number of starts; from them Ph = P·n. |
| φ, ρ′ | Lead angle of the thread and effective friction angle of the pairing. |
| d_L, μ_L | Friction diameter and coefficient of the axial support. |
| T, η | Required torque and efficiency of the screw. |
Choose the inputs correctly
Pitch diameter d₂, pitch P and number of starts n, the axial force F, the nut material with its lubrication state (or a custom friction angle), and optionally the friction diameter and coefficient of the thrust face and the speed.
How to use the calculator
Enter the pitch diameter of the thread – for Tr 20×4 that is 18 mm, not 20 mm. Add the pitch and number of starts, which give the travel per revolution, and the axial force. Then choose the nut material and lubrication state, which sets the friction angle; if you know the value, enter it directly instead. Optionally add the axial support and the speed to include thrust friction, feed rate and drive power.
Worked example
A Tr 20×4 screw (d₂ = 18 mm) at 10 kN axial force with a lubricated bronze nut (ρ′ = 6°) has φ = 4.05° and is therefore self-locking. Raising requires 15.94 N·m at 39.9 % efficiency; lowering requires 3.07 N·m to be applied, because the load does not come down by itself.
Which value decides what?
The most informative output is the comparison of lead angle and friction angle: it decides self-locking and at the same time fixes the efficiency, since both follow from the same two angles. The raising torque is the sizing value for the drive; the lowering torque says whether a brake is needed. Reporting the thread and thrust shares separately shows where an improvement pays off most – often it is the thrust bearing rather than the thread.
Diameters and pitch in millimetres, force in newtons, angles in degrees, torque in newton metres, speed in min⁻¹. The efficiency is reported as a percentage.
Typical applications
Sizing screw presses, jacks, lifting tables, clamping devices and vices, determining the hand or motor effort at a handwheel or drive, judging whether a load is held without a brake, and comparing a self-locking trapezoidal screw against a fast multi-start design.
Assumptions, limits and common mistakes
Roloff/Matek, Maschinenelemente Formelsammlung, 15th edition, section 8 "Bewegungsschrauben" (no. 72 torque, no. 81 efficiency and self-locking); reference values for ρ′ by nut material and lubrication from the same source. Only torque, efficiency and self-locking are calculated – flank pressure, buckling, combined stress and wear must be verified separately.
Common mistake: Do not enter the nominal diameter instead of the pitch diameter – on trapezoidal threads d₂ is half a pitch below it. On multi-start threads give the pitch and the number of starts separately, not their product: the travel per revolution is P·n. And do not forget the thrust friction if the screw is supported on a plain face – it can match the thread torque and then halves the efficiency.
Frequently asked questions
Why is a self-locking screw never better than 50 %?
Because both statements are the same inequality. Self-locking means φ < ρ′, and under that condition the efficiency tan φ/tan(φ + ρ′) reaches at most just under 0.5. Wanting high efficiency means giving up self-locking and providing a brake or a self-locking gear stage instead – which is why ball screws at over 90 % efficiency always need a holding brake.
Why is the lowering torque sometimes negative?
A negative value means the load does not drive the screw back by itself: the thread friction outweighs the contribution of the lead. The screw then has to be driven actively in the lowering direction. On a non-self-locking screw the value is positive and states the torque that must be braked against so the load does not run away.
What is the difference between ρ′ and the friction coefficient?
The friction angle ρ′ already contains the correction for the thread's flank angle. Because the flank is inclined, at a given axial force it presses harder than a flat surface would, so the effective friction is greater than the bare coefficient suggests. That is why the literature tabulates ρ′ directly rather than μ.
What does a multi-start thread achieve?
At the same pitch it multiplies the travel per revolution and therefore the lead angle. Efficiency rises considerably and so does the travel speed – but self-locking is usually lost, because φ rises above ρ′. The calculator shows both at once, so the trade-off can be read directly.
Is this calculation enough to size the screw?
No, it answers only the drive side. The flank pressure against the permissible value for the nut material, the buckling safety of the compressively loaded spindle, the combined compressive and torsional stress and the wear life of the nut all have to be verified as well.