Lead Screw Torque and Force Calculator
Calculate drive torque (T), axial thrust (F), lead angle, and self-locking conditions per Shigley Ch. 8.
Lead Screw Torque & Thrust Configuration
Table of Contents12 Engineering Topics • Click to expand
- 1. How Lead Screw Torque Is Calculated
- 2. Efficiency Model vs. Friction-Based Model
- 3. Raising vs. Lowering Torque Mechanics
- 4. Collar & Thrust Bearing Friction Torque
- 5. Lead, Pitch, Multi-Start & Mean Diameter
- 6. Kinematic Linear Speed & Traverse Rate
- 7. Thread Friction Coefficients & Material Pairs
- 8. Self-Locking & Backdriving Stability
- 9. Square vs. Acme & Metric Trapezoidal Threads
- 10. Complete Motor Sizing: Inertia & Losses
- 11. Step-by-Step Engineering Worked Examples
- 12. Engineering References & Authoritative Standards
1. How Lead Screw Torque Is Calculated
A lead screw (also known as a power screw or translation screw) converts rotational motion into linear thrust by sliding contact along helical thread flanks. The input torque required to overcome external axial resistance depends on load magnitude, screw lead, thread geometry, and friction.
Figure 1: Classical inclined plane unwrapped thread free-body diagram (Shigley Ch. 8). When raising, driving force F_R overcomes both the incline gravity component and friction. When lowering, if friction angle φ > λ, positive torque is required (self-locking); if λ > φ, the screw backdrives.
Overall Efficiency-Based Driving Torque
T = (F · L) / (2π · η)T- Required input driving torque (N·m or lbf·in)
F- Axial thrust force resisted by the screw (N or lbf)
L- Screw lead — axial advancement distance per full revolution (m or in)
η- Overall mechanical efficiency of the screw (0 to 1)
When manufacturer catalog data specifies an efficiency rating η, this formula provides the direct input torque. The inverse yields axial thrust: F = 2π · η · T / L.
2. Efficiency Model vs. Friction-Based Model
Mechanical engineers employ two complementary analytical methods to evaluate power screws:
| Analytical Property | Efficiency Model | Friction-Based Model (Shigley Ch. 8) |
|---|---|---|
| Governing Formula | T = F·L / (2πη) | T_R = (W·d_m/2) × [(L + πμd_m sec α) / (πd_m − μL sec α)] + T_c |
| Required Inputs | Force F, Lead L, Efficiency η | Load W, Lead L, Mean Dia d_m, Friction μ, Thread Angle α, Collar T_c |
| Thread Angle Dependency | Lumped into η | Explicitly modeled via sec α (wedging factor) |
| Lowering & Backdriving | Not calculated | Evaluates exact lowering torque T_L and self-locking margin |
| Typical Application | Catalog ball screws & precision lead screws | Custom Acme, Trapezoidal, and Square machine screw jacks |
3. Raising vs. Lowering Torque Mechanics
By unwrapping a helical thread into an inclined plane of angle λ = arctan(L / (π d_m)), static equilibrium reveals distinct equations for raising (driving against the load) versus lowering (moving with the load):
Raising Torque (T_R)
Must overcome both the axial force inclined plane resistance and the friction opposing forward rotation:
Always positive and represents peak continuous operating thread torque.
Lowering Torque (T_L)
Friction acts in reverse, resisting the natural tendency of the axial force to slide down the incline:
If T_L < 0, the load will backdrive unless an external holding brake is engaged.
4. Collar & Thrust Bearing Friction Torque (T_c)
When a power screw pushes against an axial load, the reaction thrust must be absorbed by a thrust washer, shaft shoulder, or bearing. This creates a parasitic collar friction torque:
Figure 4: Thrust collar bearing mechanics. Axial load (W) reacting against the stationary frame generates collar friction torque (T_c). Replacing flat sliding collars with rolling thrust bearings significantly reduces motor torque requirements.
Collar Friction Torque Formula
T_c = (W · μ_c · d_c) / 2T_c- Parasitic collar / thrust bearing friction torque (N·m)
W- Axial thrust load (N)
μ_c- Coefficient of friction at the thrust collar (≈ 0.08–0.15 for washers, ≈ 0.002 for rolling thrust bearings)
d_c- Mean collar friction diameter: d_c = (d_outer + d_inner) / 2
Replacing a plain friction washer with a rolling element thrust bearing reduces collar friction torque by 95%+, greatly improving drive efficiency.
To size rolling thrust bearings supporting power screw reaction forces, use our Bearing Load Calculator → and verify fatigue rating life with the Bearing Life Calculator (ISO 281) →.
5. Lead, Pitch, Multi-Start & Mean Diameter
Geometry definitions are fundamental to avoiding calculation errors:
| Parameter | Symbol | Mathematical Definition | Physical Description |
|---|---|---|---|
| Pitch | p | — | Axial distance between adjacent thread crests. |
| Number of Starts | n_s | Integer (1, 2, 3, 4) | Number of independent helical thread ridges wrapped around the cylinder. |
| Lead | L | L = p × n_s | Axial linear advancement per 360° rotation. Used in all torque formulas. |
| Mean Diameter | d_m | d_m = d − p/2 | Effective pitch diameter where thread contact normal forces act. |
| Helix Angle | λ | λ = arctan(L / (π d_m)) | Inclination angle of the unwrapped thread helix. |
6. Kinematic Linear Speed & Traverse Rate
Linear Speed Kinematic Relationship
v = (L · N) / 60v- Linear velocity of the nut (mm/s or in/s)
L- Screw lead (mm/rev or in/rev)
N- Rotational shaft speed (RPM)
Kinematic linear speed depends solely on lead and RPM. It does not dictate required torque.
7. Thread Friction Coefficients & Material Pairs
Friction between the screw and nut governs mechanical efficiency and self-locking stability. Typical engineering friction coefficients from Machinery's Handbook:
| Screw & Nut Material Combination | μ (Dry Running) | μ (Lubricated / Grease) | Typical Engineering Application |
|---|---|---|---|
| Steel Screw on Bronze Nut (SAE 660 / CuSn12) | 0.15 – 0.20 | 0.10 – 0.12 | Standard machine tool lead screws, high wear resistance. |
| Steel Screw on Polymer Nut (Delrin / PTFE / PEEK) | 0.10 – 0.15 | 0.06 – 0.10 | 3D printers, medical actuators, quiet maintenance-free operation. |
| Steel Screw on Cast Iron Nut | 0.20 – 0.25 | 0.12 – 0.15 | Heavy manual presses and workshop vise spindles. |
| Steel Screw on Steel Nut | 0.20 – 0.30 | 0.14 – 0.18 | Clamping bolts (prone to galling under high continuous cycling). |
8. Self-Locking & Backdriving Stability
A power screw is defined as self-locking if the axial load cannot cause the screw to rotate backwards when external driving torque is removed.
Figure 3: Mechanical efficiency (η) versus helix angle (λ) for μ = 0.12. A power screw is strictly self-locking only when η < 50% (λ ≤ φ = arctan μ). Screws with efficiency > 50% will backdrive unless held by a motor holding torque or brake.
Self-Locking Stability Condition
μ_eff ≥ tan(λ) ⟺ ϕ ≥ λ, where ϕ = arctan(μ_eff)μ_eff- Effective friction coefficient: μ_eff = μ / cos(α)
λ- Helix angle = arctan(L / (π · d_m))
ϕ- Friction angle
When φ ≥ λ, lowering torque TL is positive, requiring active torque to descend. When φ < λ, the screw is backdriveable.
Theoretical static self-locking (φ ≥ λ) can degrade rapidly under machinery vibration, lubricant thinning at elevated temperatures, or shock loads. In vertical hoist applications, always incorporate a secondary mechanical brake or safety holding pawl.
9. Square vs. Acme & Metric Trapezoidal Threads
Different thread profiles introduce varying levels of normal force wedging:
Figure 2: Power screw thread profile geometry. Square threads offer the lowest friction factor (α = 0°), Acme/Trapezoidal threads introduce a small flank wedging factor (1/cos α) for easier machining and split-nut wear compensation, while Ball Screws replace sliding friction with rolling elements.
The trapezoidal flank angle α increases the normal force between thread faces by sec α = 1 / cos α. For standard threads:
- Square Threads (α = 0°): Maximum theoretical efficiency (sec 0° = 1.0), but difficult to machine and cannot compensate for flank wear.
- Acme Threads (ASME B1.5, 2α = 29°, α = 14.5°): Widely used in North America. Wedging factor sec 14.5° ≈ 1.033 adds 3.3% friction drag while drastically increasing root shear strength.
- ISO Metric Trapezoidal (ISO 2901 / DIN 103, 2α = 30°, α = 15°): Standard in Europe and worldwide metric machinery. Wedging factor sec 15° ≈ 1.035.
10. Complete Motor Sizing: Inertia & Losses
The thread torque T_R is only one component of total motor shaft torque. When selecting servo or stepper motors, evaluate:
T_motor = (T_R + T_c) / η_gearbox + J_total × α_accel
- T_c: Collar / thrust bearing parasitic friction torque
- η_gearbox: Planetary or timing belt reducer efficiency (typically 85%–95%)
- J_total: Reflected rotational inertia of motor rotor, screw shaft, coupling, and linear payload mass m(L / 2π)²
- α_accel: Angular acceleration rate rad/s²
11. Step-by-Step Engineering Worked Examples
Example 1 — Efficiency-Based Sizing
An actuator requires moving an axial force F = 2,000 N with a 5 mm/rev lead screw rated at 40% mechanical efficiency (η = 0.40).
1. Lead in meters: L = 0.005 m
2. Driving Torque: T = (2000 N × 0.005 m) / (2π × 0.40) = 10 / 2.5133 = 3.98 N·m (3,979 N·mm)
Example 2 — Friction-Based Raising Torque (Tr 24x5 Trapezoidal)
A metric Trapezoidal screw (d = 24 mm, p = 5 mm, d_m = 21.5 mm, α = 15°) lifts a load W = 10,000 N with bronze nut friction μ = 0.12 and thrust collar friction (d_c = 30 mm, μ_c = 0.08).
1. Effective friction: μ_eff = 0.12 / cos(15°) = 0.12 / 0.9659 = 0.1242
2. Thread raising torque: T_thread = (10000 × 21.5 / 2) × [5 + π(0.1242)(21.5)] / [π(21.5) − (0.1242)(5)] = 107,500 × (13.39 / 66.92) = 21.51 N·m
3. Collar friction torque: T_c = 10000 × 0.08 × 30 / 2 = 12,000 N·mm = 12.00 N·m
4. Total Required Driving Torque: T_total = 21.51 + 12.00 = 33.51 N·m
12. Engineering References & Authoritative Standards
Standard Specifications & Primary Research Literature
- ASME/ANSI B1.5:2014. Acme Screw Threads. American Society of Mechanical Engineers. [ASME B1.5 Standard Portal]
- ISO 2901:2016. ISO metric trapezoidal screw threads — Basic profile and maximum material profiles. International Organization for Standardization.
- ISO 2904:1977. ISO metric trapezoidal screw threads — Basic dimensions. International Organization for Standardization.
- DIN 103. ISO Metric Trapezoidal Screw Threads; Coarse and Fine Pitches. Deutsches Institut für Normung.
- Budynas, R. G., & Nisbett, J. K. (2020). Shigley's Mechanical Engineering Design (11th ed.). McGraw-Hill Education. Chapter 8: Screws, Fasteners, and the Design of Nonpermanent Joints.
- Oberg, E., Jones, F. D., Horton, H. L., & Ryffel, H. H. (2020). Machinery's Handbook (31st ed.). Industrial Press. Power screws, lead screw torque, and thread profiles.
- Ham, C. W., & Ryan, D. G. (1932). An Investigation of the Efficiency and Wear of Lead Screws. University of Illinois Engineering Experiment Station Bulletin No. 247. [Illinois Digital Archive]
- Wikipedia Foundation. Leadscrew Mechanics & Power Screws. [Wikipedia: Leadscrew] and [Trapezoidal Thread Form] .
This calculator provides theoretical thread torque, linear speed, and self-locking estimations based on standard classical mechanics (Shigley Ch. 8, ASME B1.5, ISO 2901). Real-world power screw performance is influenced by manufacturing tolerances, thermal expansion, misalignment, grease degradation, dynamic shock, and inertia. In safety-critical load holding applications (vertical hoists, elevators, presses), never rely solely on theoretical self-locking; always implement an independent mechanical brake.