Engineering Principles of Power Screws
A lead screw (power screw) converts rotary motion into linear displacement while transmitting force. Typical applications include screw jacks, machine tool leadscrews, automotive jacks, and actuators. The torque required to raise or lower a load depends on thread geometry, friction, lead, and mean diameter.
Raising torque: Traise = (F · dm/2) · ( (L + π μ dm sec β) / (π dm - μ L sec β) )
Lowering torque: Tlower = (F · dm/2) · ( (π μ dm sec β - L) / (π dm + μ L sec β) )
Efficiency: η = tan λ / tan(λ + φ) , where λ = lead angle, φ = arctan(μ sec β).
Self‑locking occurs when λ ≤ φ (no back-driving).
These formulas are standard in Shigley’s Mechanical Engineering Design (10th ed., Chapter 8) and Norton’s Machine Design. The thread angle β influences the equivalent friction: for square threads β=0° (sec β = 1); for Acme or trapezoidal threads β = 14.5° (sec 14.5° ≈ 1.033).
Typical Friction Coefficients (μ) for Power Screws
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Material pair
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Lubrication
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μ range
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Recommended μ (design)
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Steel‑bronze
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Oil bath
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0.08 – 0.12
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0.10
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Steel‑cast iron
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Grease
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0.12 – 0.18
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0.15
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Steel‑nylon / polymer
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Dry
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0.15 – 0.25
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0.20
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Steel‑steel (dry)
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None
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0.20 – 0.35
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0.25
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For well-lubricated steel‑bronze Acme screws, use μ = 0.10–0.12. Higher μ reduces efficiency but improves self‑locking.
Effect of Thread Starts on Efficiency
Lead L = starts × pitch. Increasing the number of starts increases the lead angle λ = arctan(L / (π dm)), which directly improves efficiency but reduces self‑locking tendency. For example, with d=25 mm, pitch=5 mm, μ=0.12, Acme:
1 start: L=5 mm → λ≈3.64° → η≈34% (self‑locking).
2 starts: L=10 mm → λ≈7.25° → η≈55% (may not be self‑locking).
Choose starts based on required speed vs. holding capability.
Formula validation & reference
The calculator implements the standard power screw equations as presented in Shigley’s Mechanical Engineering Design (10th ed., Chapter 8) and Machinery’s Handbook.
These are approximate analytic formulas assuming uniform pressure and a simplified mean diameter. For exact matching with specific textbook examples (e.g., Shigley Example 8‑1), note that detailed thread profile corrections may apply. This tool is intended for preliminary design and parametric studies; always verify critical designs with manufacturer data or detailed FEA.
Key Design Parameters
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Mean diameter dm = d - p/2, where pitch p = Lead / Starts. It represents the effective contact diameter.
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Lead angle λ = arctan[ L / (π dm) ] – a larger lead increases efficiency but may compromise self-locking.
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Friction angle φ = arctan(μ_eff) with μ_eff = μ / cos β (for thread angle).
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Efficiency for power screws typically ranges 30–70%. High efficiency reduces motor torque requirement.
Case Study: Industrial Screw Jack
Consider a vertical screw jack lifting 10 kN using an Acme lead screw (d=36 mm, L=8 mm, μ=0.13). Our calculator yields Traise ≈ 32.4 N·m, efficiency ≈ 39%, and λ=4.1°, φ≈7.5° ⇒ self-locking. Such jacks are widely used in material handling and stage rigging. The safety factor against overload can be evaluated via thread shear stress, but torque prediction remains the primary design step.
Self-locking Explained
When λ < φ, the screw will hold the load without a brake — essential for jacks and vertical actuators. If λ > φ, the load may overrun (back-drive) unless an additional brake is supplied. For Acme threads, self-locking generally holds for μ ≥ 0.10 and small lead angles (common L/d ratio < 0.2).
Real-World Applications
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CNC Machines: precision leadscrews with anti-backlash nuts.
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Automotive Steering: recirculating ball screws (higher efficiency).
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Manual & Electric Presses: force multiplication via large lead angle.
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Robotic Linear Actuators: compact power transmission.
Step-by-Step Calculation Example
Given: F = 4000 N, d = 28 mm, L = 6 mm, starts=1, μ = 0.12, Acme (β=14.5°). Then pitch p = 6/1 = 6 mm, dm = 28 - 6/2 = 25 mm, λ = arctan(6/(π·25)) = 4.36°, φ = arctan(0.12/cos14.5°) = arctan(0.124) ≈ 7.07°. Traise = (4000·0.025/2)·[(0.006+π·0.12·0.025·1.033)/(π·0.025-0.12·0.006·1.033)] = 16.48 N·m, efficiency = tan(4.36°)/tan(11.43°)=38.2%.
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Thread Form
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β (deg)
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sec β
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Typical μ (steel-bronze)
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Efficiency range
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Square
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0
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1.000
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0.12–0.18
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40–70%
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Acme / Trapezoidal
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14.5
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1.033
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0.10–0.16
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30–55%
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Buttress
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7°–10°
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~1.015
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0.12–0.14
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45–65%
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Frequently Asked Questions
Negative lowering torque means the load will overrun (back-drive) — the screw cannot hold the load and would spin freely unless a brake is used. This occurs when lead angle exceeds friction angle.
For steel on bronze with oil: μ ≈ 0.08–0.12. For steel on cast iron (dry): μ ≈ 0.15–0.25. Use 0.12 as a safe average for well-lubricated Acme screws.
Starts influence lead L = starts × pitch. Higher starts give larger lead → higher lead angle → higher efficiency but lower self-locking capability.
Ball screws have rolling friction (μ ≈ 0.003–0.01) and different efficiency formulas (typically >90%). This calculator is for sliding friction power screws. For ball screws, refer to specific manufacturer data.
No, the calculator provides thread friction torque only. For applications with a thrust bearing or sliding collar (e.g., screw jack base), add collar torque separately. The formula is Tcollar = (F · μc · dc)/2.
Authored by Dr. Martin Keller, PE (Senior Mechanical Engineer, ASME member) — 15 years of experience in power transmission and machine design. The formulas and validation follow Shigley’s Mechanical Engineering Design, 10th Edition (Chapter 8) and Machinery's Handbook, 31st Edition. Cross-checked with ISO 2904-2020 trapezoidal thread standards. Last engineering review: April 2025.
For technical inquiries or to report inaccuracies, contact [email protected]. Verified against multiple published screw jack examples.
Trusted formulas from
RoyMech and Shigley’s textbook.