Worm Gear Ratio and Efficiency Calculator
Worm Gear Ratio and Efficiency Calculator
Ratio, lead angle, sliding velocity and efficiency for a worm pair — with the heat that low efficiency implies, the thermal rating that heat forces, and the reason self-locking must never be the only thing holding a load.
Worm gear ratio and efficiency
A double-start 4 mm axial module worm of 50 mm pitch diameter driving a 40-tooth wheel, 5.5 kW at 1,450 rev/min, friction from the published curve
Ratio in one stage, and the efficiency that costs
- z_worm
- number of starts or threads on the worm, 1 to 6. The ratio divides by it and the lead angle multiplies by it, which is the whole trade
- γ
- lead angle, measured from the plane of rotation of the worm. Everything depends on it
- d₁
- worm pitch diameter — the free parameter. A slimmer worm has a steeper lead angle and runs more efficiently, at the cost of a more slender and less stiff worm shaft
- μ
- coefficient of friction at the mesh. Strongly dependent on the SLIDING velocity, not the rotational speed, and that dependence is what breaks self-locking
- η
- driving efficiency, worm to wheel. Derived from η = tan γ/tan(γ + φ) with tan φ = μ, generalised to a non-zero normal pressure angle. At α_n = 0 it reduces to tan γ(1−μ·tan γ)/(tan γ+μ), which is the screw-thread form most sources print
- η’
- BACK-DRIVING efficiency, wheel to worm: the same expression with tan and cot exchanged. Zero or negative means the wheel cannot turn the worm under static friction, which is what self-locking means and all it means
Worked example
A double-start 4 mm axial module worm of 50 mm pitch diameter driving a 40-tooth wheel, 5.5 kW at 1,450 rev/min, friction from the published curve
The ratio first, because it is the reason anybody uses a worm: z_wheel/z_worm = 40/2 = 20:1, in a single stage, at a right angle, in a housing the size of a fist. A spur or helical train needs two stages to do that and a bevel pair cannot do it at all above about 6:1
The lead is z_worm·π·m_x = 2·π·4 = 25.1327 mm, so the lead angle is arctan(lead/πd₁) = arctan(z_worm·m_x/d₁) = arctan(8/50) = 9.090°. The diameter quotient q = d₁/m_x is 12.50, which is mid-range for a catalogue worm
Now the number the friction depends on, and it is NOT the rotational speed. The worm's peripheral velocity is πd₁n₁/60000 = 3.796 m/s, and the SLIDING velocity at the mesh is that divided by cos γ = 3.844 m/s. At that speed the published curve for a case-hardened steel worm on a phosphor-bronze wheel gives μ = 0.0264
THE EFFICIENCY. η = (cos α_n − μ·tan γ)/(cos α_n + μ·cot γ) = (0.9397 − 0.00423)/(0.9397 + 0.16518) = 84.67 per cent. Note what the formula most sources print would have said: the zero-pressure-angle screw form tan γ(1−μ·tan γ)/(tan γ+μ) gives 85.46 per cent, which is 0.79 points high. It is the α_n = 0 case, and a real 20° normal pressure angle needs a larger normal force for the same tangential force, so more friction
What that costs in torque and heat. 5.5 kW at 1,450 rev/min is 36.221 N·m at the worm. A lossless 20:1 drive would give 724.4 N·m at the wheel; at 84.7 per cent you get 613.4 N·m. The missing 15.3 per cent is 843 W of heat, continuously, and about 675 W of it appears in the worm rather than the wheel
Now the self-locking question, which is what most people come here for. Back-driving efficiency is the same expression with tan and cot exchanged: 82.1 per cent — positive, so this pair back-drives freely. The threshold is tan γ = μ/cos α_n, which here means γ below 1.611°. Your 9.09° is nowhere near it. Note that the condition everyone quotes, tan γ < μ, would give 1.514° — the pressure angle moves the threshold up by 6.4 per cent, and SDP/SI states the condition in the equivalent form (cos 20°·sin γ − μ·cos γ) ≤ 0
And the warning that matters more than any of the arithmetic. To make this pair self-lock you would need a lead angle under 1.61°, which means a single start on a fat worm — and at that lead angle the forward efficiency would be about 47 per cent. The self-locking lead angles ARE the inefficient ones; that is one fact, not two. And even then it would not be a holding device: μ rises to 0.12 at rest and falls to 0.044 at 1 m/s, so a pair that locks while still can back-drive as soon as anything moves. Machinery's Handbook: “if irreversibility is desired, it is recommended that some form of brake be employed”
Self-locking is not a holding device
| The position | The source, in its own words |
|---|---|
| Self-locking is a STATIC property only | Design World: “no gear assembly is absolutely self-locking. Rather, worm gears’ locking behavior is only exhibited from a static (stopped) state.” A worm box cannot slow a moving load — the axis’ own inertia keeps it turning after the power is cut, and the mesh only locks once it has already stopped |
| Vibration breaks it | Machine Design: “if this self-locking reducer is subjected to shock and vibration, the friction coefficient between worm and gear may suddenly drop”, and once back-driving starts it “usually continues because the friction coefficient decreases with increasing speed”. The friction table on this page shows exactly that: μ falls from 0.12 at rest to 0.044 at 1 m/s |
| Every published analysis is a static one | Machine Design again: “all theoretical analyses of self-locking worm gears deal with static conditions”, and “such vibration-free applications are rare”. So the calculation on this page — and every other page’s — answers a question narrower than the one being asked of it |
| Machinery’s Handbook has said so for decades | “It is usually impractical to design irreversible worm gearing with any security. If irreversibility is desired, it is recommended that some form of brake be employed.” This is the oldest and bluntest statement of the position |
| So: add a brake | Machine Design’s instruction, verbatim: “if safety is a potential issue, always add an adequate braking device, whether or not vibration and shock are expected.” Note “whether or not”. The working arrangement Design World describes is a brake to STOP the load and a self-locking box to HOLD it once stopped — the two do different jobs and you need both |
Efficiency and back-driving against lead angle, at μ = 0.05
| Lead angle γ | Efficiency (with α_n = 20°) | … by the screw formula | Overstatement (points) | Back-driving efficiency | Verdict |
|---|---|---|---|---|---|
| 2.0° | 39.6% | 41.0% | 1.50 | -52.3% | self-locking |
| 3.0° | 49.5% | 51.0% | 1.56 | -1.5% | self-locking |
| 5.0° | 61.9% | 63.4% | 1.46 | 39.0% | back-drives |
| 7.5° | 70.7% | 72.0% | 1.28 | 59.2% | back-drives |
| 10.0° | 76.1% | 77.2% | 1.12 | 69.2% | back-drives |
| 15.0° | 82.2% | 83.1% | 0.90 | 79.0% | back-drives |
| 20.0° | 85.6% | 86.3% | 0.77 | 83.8% | back-drives |
| 25.0° | 87.5% | 88.2% | 0.68 | 86.4% | back-drives |
| 30.0° | 88.7% | 89.4% | 0.62 | 88.1% | back-drives |
Friction against sliding velocity, and what it does to both answers
| Sliding velocity (m/s) | μ | Self-locking threshold lead angle | Efficiency at γ = 10° | Efficiency at γ = 20° |
|---|---|---|---|---|
| 0.001 | 0.120 | 7.277° | 56.7% | 70.6% |
| 0.100 | 0.080 | 4.866° | 66.4% | 78.5% |
| 1.000 | 0.044 | 2.681° | 78.4% | 87.1% |
| 5.000 | 0.023 | 1.402° | 87.4% | 92.9% |
| 10.000 | 0.018 | 1.097° | 89.9% | 94.3% |
| 20.000 | 0.016 | 0.975° | 90.9% | 94.9% |
A worm box has TWO ratings and the smaller one governs
| Rating | What it limits | What sets it |
|---|---|---|
| Mechanical rating | Torque and speed the teeth, shafts and bearings can carry | Set by tooth bending and surface durability, and by the shaft and bearing capacities |
| Thermal rating | Continuous power the box can shed as HEAT without the oil overheating | Set by the case’s surface area, the airflow over it, the ambient temperature and the oil. AGMA/ISO 14179 refers it to a stated ambient and a stated sump temperature — typically 25 °C ambient and a 95 °C sump limit |
| Why worm boxes are usually thermally limited | Because the loss is enormous compared with any other gear type | A worm mesh at 45 per cent efficiency turns 55 per cent of the input into heat. A helical pair at 96 per cent turns 4 per cent into heat. That is a factor of fourteen in the cooling problem for the same transmitted power, from the same size of box |
| Where the heat appears | Mostly in the WORM, not the wheel | Roughly 80 per cent of the loss appears at the worm, because the worm’s surface slides continuously against the same wheel teeth while each wheel tooth only meets the worm once per revolution. So the worm shaft and its bearings run hotter than the sump, and that is where a thermally overloaded box fails first |
| What a service factor does NOT do | It does not provide thermal margin | In TANHON’s words: applying a service factor “does not” give thermal margin, and a box “might have a 13 HP mechanical rating but only a 10 HP thermal rating at 875 RPM input”. The two ratings are independent limits and the smaller one governs. Always look the thermal rating up separately |
A huge ratio in one stage, the heat that costs, and why self-locking is not a brake
A worm drive gives a huge ratio in one stage, and that is the whole reason to accept everything else about it. The ratio is z_wheel/z_worm — the number of WHEEL teeth over the number of worm starts — so a single-start worm on a 40-tooth wheel is 40:1 in one mesh, at a right angle, in a housing you can hold. Nothing else does that: a spur or helical stage runs out of sense above about 7:1 and a bevel pair above about 6:1, so both need two stages. The worm’s geometry follows from three numbers — the starts, the axial module and the worm’s own pitch diameter — and the pitch diameter is the free one: tan γ = z_worm·m_x/d₁, so a slimmer worm at the same module and starts has a steeper lead angle.
The honest answer contradicts the folklore: worm drives run 40 to 90 per cent efficient, and a single-start worm at a high ratio can be under 50. The efficiency is η = (cos α_n − μ·tan γ)/(cos α_n + μ·cot γ), which comes from η = tan γ/tan(γ + φ) with tan φ = μ, generalised to a real pressure angle. Most sources print the α_n = 0 version, tan γ(1 − μ·tan γ)/(tan γ + μ), which is a square-thread screw and OVERSTATES the efficiency — by more than a point and a half at a 5° lead angle, because a 20° normal pressure angle needs a larger normal force for the same tangential force and therefore a larger friction force. Both are on the page so you can see the size of the difference.
That efficiency is a heat problem, and it is why a worm box has two ratings. A 45 per cent drive turns 55 per cent of its input into heat, continuously; a helical pair at 96 per cent turns 4 per cent into heat. That is a factor of fourteen in the cooling job for the same transmitted power. So worm gearbox catalogues print a MECHANICAL rating, set by the teeth and shafts, and a separate THERMAL rating, set by what the case can shed — referred by AGMA/ISO 14179 to a stated ambient and sump temperature — and the smaller of the two governs. Two traps: a service factor does not provide thermal margin, and about eighty per cent of the heat appears in the WORM rather than the wheel, so the worm shaft, bearings and seals run hotter than the sump.
Self-locking is not a design feature you can rely on. It needs tan γ < μ/cos α_n — a lead angle of roughly five degrees or less at ordinary friction — and the condition everybody quotes, tan γ < μ, leaves out the pressure angle and puts the threshold about six per cent low. But the arithmetic is the least of it. VIBRATION BREAKS IT. The friction coefficient at the mesh falls from about 0.12 at rest to 0.044 at one metre per second of sliding, so the threshold falls with it; Machine Design reports that under shock and vibration “the friction coefficient between worm and gear may suddenly drop” and that once back-driving starts it “usually continues because the friction coefficient decreases with increasing speed”. Design World adds that “no gear assembly is absolutely self-locking” and that the behaviour “is only exhibited from a static (stopped) state” — so a worm box cannot even slow a moving load. Machinery's Handbook has said for decades that “it is usually impractical to design irreversible worm gearing with any security. If irreversibility is desired, it is recommended that some form of brake be employed.” Never use a worm’s self-locking as the only means of holding a load.
And the two facts are one fact. Look at the efficiency curve against lead angle with the self-locking region shaded: the lead angles that self-lock are the lead angles that waste more than half the power. That is not a coincidence — it is the same μ appearing in both directions. You cannot buy a worm that both holds its load and runs efficiently, and anyone who believes they have one has not measured it hot. For the ratio and torque through the rest of the train, see the coupling torque and selection calculator; for the service factor your machine class needs, the drive service factor calculator; and for a right-angle drive at a ratio a worm does not suit, the bevel gear geometry calculator.
Frequently asked questions
What is the efficiency of a worm gear?
Between about 40 and 90 per cent, and which end you land on depends almost entirely on the LEAD ANGLE. η = (cos α_n − μ·tan γ)/(cos α_n + μ·cot γ): at a 20° lead angle and μ = 0.05 that is about 89 per cent, and at 3° it is about 41 per cent. Since tan γ = z_worm·m_x/d₁, the levers are more worm starts and a slimmer worm. A single-start worm at a high ratio is the worst case and can be under 50 per cent.
Can a worm gear be used to hold a load without a brake?
No. Not on a hoist, a lift, a gate, a jack or anything that can fall. A worm pair with a lead angle below arctan(μ/cos α_n) will not be back-driven while it is stationary and while the friction is what you assumed — and both of those fail in service. The friction coefficient drops by a factor of seven between rest and 20 m/s of sliding, and Machine Design reports that under shock and vibration it “may suddenly drop”, after which back-driving “usually continues because the friction coefficient decreases with increasing speed”. Machinery’s Handbook’s advice is to fit a brake. Fit a brake.
At what lead angle is a worm gear self-locking?
When tan γ < μ/cos α_n, which for μ = 0.08 and a 20° normal pressure angle is a lead angle below 4.87°. Note the cos α_n: the condition usually quoted, tan γ < μ, gives 4.57° and is the zero-pressure-angle case. SDP/SI states the same condition as (cos 20°·sin γ − μ·cos γ) ≤ 0, which rearranges to exactly that. And because μ depends on the sliding velocity, the threshold moves: at rest it may be 7°, at one metre per second it is under 3°.
Why do worm gearboxes have a separate thermal rating?
Because the loss is enormous compared with any other gear type and the heat has to leave through the case. A worm box at 45 per cent efficiency turns 55 per cent of its input into heat continuously, where a helical pair at 96 per cent turns 4 per cent into heat — fourteen times the cooling problem for the same transmitted power. So the catalogue prints a mechanical rating from the teeth and shafts and a thermal rating from what the case can shed, referred by AGMA/ISO 14179 to a stated ambient and sump temperature, and the smaller one governs. Applying a service factor does not give thermal margin.
Does a worm drive’s efficiency depend on which way it is driven?
Yes, strongly, and that is what self-locking is. Driving efficiency (worm to wheel) is (cos α_n − μ·tan γ)/(cos α_n + μ·cot γ); back-driving efficiency (wheel to worm) is the same expression with tan and cot exchanged. At a shallow lead angle the back-driving efficiency goes NEGATIVE, meaning the wheel cannot turn the worm at all under static friction. Both numbers are on this page, because a drive that is 45 per cent efficient forwards and cannot be back-driven at all is a very different machine from one that is 89 per cent forwards and 87 per cent backwards.
How do I improve a worm drive’s efficiency?
Steepen the lead angle, which means one of three changes. More worm STARTS: going from one to two doubles tan γ at the same worm diameter, and to keep the ratio you double the wheel teeth. A SLIMMER worm: tan γ = z_worm·m_x/d₁, so reducing d₁ at the same module steepens it — at the cost of a less stiff worm shaft. Or a lower μ: the right lubricant, a harder and better-finished worm, and a bronze wheel rather than iron. The friction curve also falls with sliding speed, so a worm drive is often more efficient at speed than at a crawl.
What is the diameter quotient q of a worm?
q = d₁/m_x, the worm’s pitch diameter divided by its axial module, and it is how catalogues parametrise the one free choice in a worm’s geometry. Typical values run from about 7 to about 15. It matters because tan γ = z_worm/q — so the lead angle, and therefore the efficiency and the self-locking behaviour, is set by the number of starts and q alone, independently of the module. A low q is a slim, efficient, flexible worm; a high q is a fat, stiff, inefficient one.
Related calculators
References
- Machinery’s Handbook, on irreversible worm gearing, quoted in Machine Design’s Self-locking worm gears: fact or fiction?: “It is usually impractical to design irreversible worm gearing with any security. If irreversibility is desired, it is recommended that some form of brake be employed.” This is the oldest and bluntest statement of the position this page takes.
- Machine Design, Self-locking worm gears: fact or fiction?. The source for the mechanism by which self-locking fails: “if this self-locking reducer is subjected to shock and vibration, the friction coefficient between worm and gear may suddenly drop”, and once back-driving starts it “usually continues because the friction coefficient decreases with increasing speed”. Also for the observation that “all theoretical analyses of self-locking worm gears deal with static conditions” while “such vibration-free applications are rare”, and for the instruction this page repeats: “if safety is a potential issue, always add an adequate braking device, whether or not vibration and shock are expected.”
- Design World, PSA: Self-locking gearmotors aren’t brakes. The source for the distinction that makes the danger concrete: “no gear assembly is absolutely self-locking. Rather, worm gears’ locking behavior is only exhibited from a static (stopped) state” — a worm box cannot slow a moving load, because the axis’ own inertia keeps it turning after the power is cut and the mesh only locks once it has already stopped. The working arrangement it describes is a brake to stop the load and a self-locking box to hold it afterwards.
- RoyMech, Worm Gears: Design, Formula, Efficiency, Strength (BS 721 & AGMA). The source for the friction curve used here — μ against sliding velocity for a case-hardened steel worm on a phosphor-bronze wheel, from 0.12 at 0.001 m/s to 0.016 at 20 m/s — and for the efficiency formula in the form that carries the normal pressure angle. Its sliding-velocity expression 0.00005236 d1n1sec γ was checked and is exactly π/60000, which is the internal consistency test this page relied on; an earlier batch on this site found a shifted column on a different RoyMech page, so nothing here was taken on trust.
- Stock Drive Products / Sterling Instrument, Elements of Metric Gear Technology (the technical section of catalogue D805). The source for two things taken as printed here: the minimum tooth count free of undercut — “for 14.5° the value of zc is 32, and for 20° it is 18” — and the contact ratio guidance, “it is good practice to maintain a contact ratio of 1.2 or greater. Under no circumstances should the ratio drop below 1.1, calculated for all tolerances at their worst case values.” Also the source for the self-locking condition in the form (cos 20° sin γ − μ cos γ) ≤ 0, which is what puts the normal pressure angle into the threshold.
- ANSI/AGMA 6034-B92, Practice for Enclosed Cylindrical Wormgear Speed Reducers and Gearmotors, and AGMA/ISO 14179-1, Gear Reducers — Thermal Capacity Based on ISO/TR 14179-1. Cited by number. These are the documents that make the thermal rating of a worm box a SEPARATE rating from its mechanical one, referred to a stated ambient temperature and a stated sump temperature.
- TANHON, Gearbox Thermal Rating: Avoiding Overheating Issues. Cited for the statement of the trap in plain words: a box “might have a 13 HP mechanical rating but only a 10 HP thermal rating at 875 RPM input”, applying a service factor “does not” provide thermal margin, and for worm boxes “always verify thermal rating explicitly”. Also for the observation that roughly 80 % of the heat appears in the worm rather than the wheel, which is why the worm shaft and its bearings are where a thermally overloaded box fails first.
- NPTEL / IIT Madras, Machine Design II, lecture 16 — worm gears, worked problems. Two published worked examples that this page reproduces: a 2-start worm of 50 mm pitch diameter with a 32-tooth, 4 mm module wheel gives a 25.12 mm lead and a 9.09° lead angle, and 2 kW at 2,950 rev/min through it at 85.9 % gives 89.0 N·m of output torque — all four reproduced here to three figures. Its second example’s stated 18.43° lead angle is consistent with a 72 mm worm pitch diameter and not with the 80 mm the text gives, and its first example’s stated centre distance disagrees with its own diameters; both are recorded rather than used.