Worm gear calculator
Calculate the geometry and efficiency of a cylindrical worm (DIN 3975): enter number of starts, worm wheel tooth count, axial module and the diameter factor q = d1/m_x, choose coefficient of friction or friction angle - the calculator returns ratio, lead angle, diameters, centre distance, driving efficiency and the self-locking check with traffic-light rating, live with every input. Load capacity and heating per DIN 3996 are not included.
Calculation
Model: cylindrical worm per DIN 3975. It computes geometry, driving efficiency and self-locking. Load capacity, heating and wear per DIN 3996 (tooth breakage, pitting, wear, thermal power limit) are not included. A tool for geometric preliminary sizing.
Results
Calculating …
Calculation in your browser, inputs go to our server only when you export or save.
Formulas and fundamentals
Geometry
The geometry of a cylindrical worm follows from a few parameters. The ratio is:
with the number of starts z1 of the worm and the tooth count z2 of the worm wheel. The worm diameter is set through the diameter factor q with the axial module m_x:
the worm wheel reference diameter and the centre distance are:
The mean lead angle γ and the lead follow from:
Efficiency and output torque
The driving efficiency (the worm drives the wheel) is:
with the friction angle ρ' = arctan(µ) and the tooth-flank coefficient of friction µ. It rises with the lead angle: small lead angles (few starts, large diameter factor) give low efficiency, multi-start worms with a small diameter factor give high efficiency. The output torque follows from:
Self-locking
Self-locking occurs when the lead angle does not exceed the friction angle:
Then the wheel cannot drive the worm - static self-locking is often used for lifting and holding tasks but always comes with an efficiency below 50 percent. The coefficient of friction µ depends strongly on the material pairing, lubrication and sliding speed; typical values range from 0.03 to 0.10.
Worked example
A two-start worm (z1 = 2) drives a worm wheel with z2 = 40 teeth, axial module m_x = 4 mm and diameter factor q = 10; the flank coefficient of friction is µ = 0.05. The ratio is i = z2/z1 = 20. From tan(γ) = z1/q = 0.2 the lead angle is γ = 11.31°.
The diameters are d1 = q·m_x = 40 mm and d2 = m_x·z2 = 160 mm, the centre distance a = (40 + 160)/2 = 100 mm. The friction angle is ρ' = arctan(0.05) = 2.86°.
The driving efficiency is η = tan(11.31°)/tan(11.31° + 2.86°) = 0.792, about 79 percent. Because γ > ρ' the gear is not self-locking. With a drive torque T1 = 20 Nm the output torque is T2 = 20·20·0.792 = 317 Nm.
Frequently asked questions
What does the diameter factor q mean for a worm gear?
The diameter factor q = d1/m_x relates the worm diameter d1 to the axial module m_x. Together with the number of starts it sets the lead angle (tan(γ) = z1/q). Common values lie roughly between 6 and 17: a small q gives large lead angles and high efficiency, a large q a stiffer, easily supported worm. Alternatively d1 can be specified directly.
When is a worm gear self-locking?
When the lead angle γ does not exceed the friction angle ρ' = arctan(µ), i.e. γ <= ρ'. Then the worm wheel cannot back-drive the worm. Self-locking arises at small lead angles (single-start worm, large diameter factor) and higher friction and always comes with low efficiency. Note: static self-locking does not reliably prevent back-driving under shock or vibration.
How does efficiency depend on the lead angle?
The driving efficiency η = tan(γ)/tan(γ + ρ') rises with the lead angle and falls with the friction angle. Large ratios need small lead angles because of the low number of starts and therefore have lower efficiency. Multi-start worms (z1 = 2 … 4) with a small diameter factor reach 80 to over 90 percent, while strongly self-locking designs are below 50 percent.
Which coefficient of friction should I use?
The tooth-flank coefficient of friction depends on the material pairing (usually a steel worm on a bronze wheel), lubrication and sliding speed. Typical design values range from 0.03 to 0.10; at low sliding speeds and poor lubrication rather higher. Alternatively the friction angle ρ' can be entered directly. For reliable values consult manufacturer data and DIN 3996.
What does this calculator not compute?
It provides geometry, efficiency and self-locking but no load capacity check. Tooth breakage, pitting, wear and above all the thermal power limit (heating) per DIN 3996 are not included - heating in particular is often the limiting factor for worm gears. The calculator is a tool for geometric preliminary sizing, not a final design.
Related tools
Gear load capacity calculation
Pro exportVerify tooth root and flank stress of a spur gear pair per DIN 3990 / ISO 6336, with the safety factors SF and SH.
Bevel Gear
Pro exportCalculate a straight bevel gear online: pitch cone angles, outer and mean pitch diameters, cone distance, virtual tooth number and tangential force with a sketch and traffic-light rating.
Gear Ratio & Transmission
Free reportGear ratio i from tooth counts, diameters or a speed pair, reflecting speed, torque and inertia through the stage.
Drive Sizing Calculator
Pro exportSize the complete drivetrain: motor, gearbox, inertias and torques in a single calculator.