The condition for self-locking
Self-locking means that an external load cannot cause the drive shaft to rotate backward, even though no brake is applied—the gearbox locks itself automatically. This occurs when the lead angle γ of the worm is less than or equal to the friction angle ρ′ at the point of tooth engagement:
Self-locking occurs when: γ ≤ ρ′, where ρ′ = arctan(μ / cos α_n)
γ = lead angle of the worm · μ = coefficient of friction in tooth engagement: for static self-locking, the static coefficient of friction μ₀ (lubricated steel/bronze, approximate value around 0.10–0.15); for dynamic self-locking, the lower coefficient of sliding friction (around 0.03–0.08) · α_n = Normal pressure angle (usually 20°)
As a rough rule of thumb, self-locking can be estimated based on the gear ratio, since a high gear ratio usually corresponds to a shallow lead angle. However, the lead angle (number of starts, frame size) and coefficient of friction remain decisive—the manufacturer’s specifications are binding. Guideline values from catalogs of single-stage worm gearboxes:
- i < 20:1 (multi-start, γ over approximately 12°) – no self-locking, reversible gear
- i ≈ 20–25:1 (γ approximately 7–13°) – Borderline range, static self-locking uncertain
- i ≈ 30–50:1 (usually single-start, γ usually 5–8°) – static self-locking is common, but not reliable under vibration
- i ≥ 60:1 (γ usually less than 5°) – pronounced static self-locking; dynamic self-locking only below about 3°, usually only when i = 80–100
For the same gear ratio, the angle varies depending on the size and number of threads (example: i = 30: approximately 6° for single-start, approximately 8.5° for double-start).
Source of reference values: Catalog “Classic Worm Gearboxes,” Technische Antriebselemente GmbH, 2022 edition, pp. GS 05–GS 06.
This leads to the rule of thumb from Unit 2: Static self-locking requires that the static efficiency (when starting from a standstill) be below approximately 0.5; between 0.5 and 0.55, it is uncertain. The operating efficiency is significantly higher: worm gearboxes that are listed as statically self-locking in the catalog can easily achieve 60–70% during operation. Dynamic self-locking requires an operating efficiency below approximately 0.5. Worm gearboxes (operating efficiency approximately 35–90%, depending on the reduction ratio) may therefore be self-locking or not; spur, planetary, and bevel gearboxes with η above 90% are never self-locking.
- Marked point 1: Sliding friction 0.05: not dynamically self-locking
- Marked point 2: Static friction 0.10: statically self-locking
- Marked point µ = 0.05 – Sliding friction 0.05: not dynamically self-locking
- Marked point 1: Static friction 0.10: statically self-locking
Description and values of the figure
The curve shows the friction angle ρ′ = arctan(µ / cos α_n) with α_n= 20° above the coefficient of friction µ. The horizontal boundary line corresponds to the assumed lead angle γ = 4°: To the left of the intersection point (µ ≈0.066), ρ′ is less than γ, and the arrangement is not self-locking; to the right of it, ρ′ is greater than or equal to γ, and the arrangement is self-locking. The point of intersection occurs at µ = tan 4° · cos 20° ≈ 0.066. At rest, the static coefficient of friction applies (0.10 in the example; to the right of the intersection point, ρ₀′ ≈ 6.07°: statically self-locking); in motion, the coefficient of sliding friction applies (0.05; to the left of the intersection point, ρ′ ≈ 3.05°: not dynamically self-locking).
| Coefficient of friction µ (stationary: static friction µ₀; motion: sliding friction µ) | Reibwinkel ρ′ (°) |
|---|---|
| 0.02 | 1.22 |
| 0.03 | 1.83 |
| 0.04 | 2.44 |
| 0.05 | 3.05 |
| 0.06 | 3.65 |
| 0.07 | 4.26 |
| 0.08 | 4.87 |
| 0.10 | 6.07 |
Static vs. dynamic
Static self-locking holds a stationary load in place. Dynamic self-locking means that the running gearbox comes to a stop on its own after the drive is shut off, even though the load continues to pull. For this to happen, the lead angle must be smaller than the angle of sliding friction—which is smaller than the angle of static friction when at rest. Dynamic self-locking therefore requires a significantly smaller lead angle and efficiency. Neither type is protected against shocks and vibration: Vibrations can set even a self-locking gear in motion.
Worked example
Given
Lead angle γ = 4°, normal pressure angle α_n = 20°; assumed: static coefficient of friction μ₀ = 0.10 (at rest), coefficient of sliding friction μ = 0.05 (in motion).
Calculation
- cos α_n = cos 20° = 0.9397
- At rest: ρ₀′ = arctan(μ₀ / cos α_n) = arctan(0.10 / 0.9397) = arctan(0.1064) ≈ 6.07° → 4° < 6.07° → statically self-locking
- Motion: ρ′ = arctan(μ / cos α_n) = arctan(0.05 / 0.9397) = arctan(0.0532) ≈ 3.05° → 4° > 3.05° → Not dynamically self-locking: Once the gearbox is in motion (due to a shock, vibration, or shutdown under load), the load continues to move.
Only a smaller lead angle of γ = 2.5° (smaller than ρ′ ≈ 3.05°) would also be dynamically self-locking according to calculations. This is consistent with manufacturer tables: 3–5° static self-locking, and below 3°, largely dynamic self-locking as well. The coefficients of friction vary widely—which is why the transition is not sharp (Question 3 in the knowledge check).
Why Self-Locking Mechanisms Are No Substitute for Brakes
Self-locking is a geometric property of the gearing—not a tested, standard-compliant safety function. Wear, heating (decreasing oil viscosity, decreasing coefficient of friction), and vibration can reduce the effective friction angle and weaken or eliminate self-locking.
Safety and Standards
Safety Note: Holding brakes, fall protection systems, and other safety functions are designed and tested based on a risk assessment and the relevant standards (e.g., DIN EN ISO 13849). This learning unit explains the fundamentals; it does not serve as a basis for such design and testing.
Typical Applications
Self-locking is useful in lifting mechanisms (in addition to a standard-compliant brake), flap mechanisms (e.g., window or cabinet doors to prevent accidental slamming), and positioning drives that are designed to maintain their position without a continuous current. In contrast, during frequent continuous operation with long runtime periods, the efficiency loss usually outweighs the benefits—in such cases, an efficient gearbox with a separate holding brake is the more economical solution.
Knowledge check
Answer all three questions, then click "Check". From 2 of 3 correct answers, the unit counts as completed. You can retry at any time.
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