Meshing external gears rotate in opposite directions. Speed magnitude is inversely proportional to tooth count.
For driver 1 and output 2: ω₂ = −ω₁ × z₁/z₂. Pitch circles have equal tangential speed.
SDP/SI · Elements of Metric Gear TechnologyHow do gears trade speed for turning force?
Turn once. Count what happens. Discover gear ratios, real-world uses, and why different jobs need different gears.
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Follow the mechanism
Our gear-ratio convention is output teeth divided by input teeth: i = z₂/z₁ = |ω₁/ω₂|.
An 18-tooth input and 36-tooth output give a 2:1 reduction: 60 RPM in produces 30 RPM out. Reversing those tooth counts gives a 1:2 ratio and 120 RPM out. The displayed examples follow this convention consistently.
KHK · Gear Trains, §2.1 / Eq. 2.1Pitch diameter equals module × tooth count. Unshifted center distance equals half the sum of pitch diameters.
Standard full-depth spur geometry: tip radius r + m; root radius r − 1.25m; base radius r cos(20°).
KHK Gears · §4.1 / Table 4.1Ideal output torque rises in the same proportion that output speed falls.
For steady, lossless transmission: |T₂| = |T₁| × z₂/z₁ and power magnitude Tω is conserved. Real losses reduce available output.
KHK · Gear Forces, §12.1 / Table 12.1Spur: simple parallel shafts.
Straight teeth mesh between gears on parallel shafts. This is the pair you can experiment with above. A simple, economical design. Ideal spur teeth produce no force along the shaft axis. Tooth engagement can be noisier and cause more vibration at high speed than a comparable helical pair.
KHK · Spur gearsHelical: smoother running.
Teeth angle around each gear. In a common parallel-shaft pair, one gear has a left-hand helix and the other a right-hand helix. Progressive tooth engagement makes a comparable drive smoother and quieter. Single-helical gears also push along the shaft axis. Bearings must handle that extra thrust. Crossed-shaft helical designs also exist.
KHK · Helical gearsBevel: intersecting shafts.
The pitch surfaces are cones. Their shaft axes intersect, commonly at 90°. A 1:1 miter pair redirects rotation without changing its speed magnitude. Transfer rotation between intersecting shafts. Spiral bevel teeth can run more smoothly than straight bevel teeth. Alignment and bearing loads matter. Spiral bevel gears are harder to manufacture. A complete differential needs more than this two-gear pair.
KHK · Bevel gearsWorm: large compact reduction.
A screw-like worm drives a worm wheel. The shafts are nonparallel and do not intersect, often with a 90° angle between their directions. A large reduction in a compact, quiet single stage. Sliding contact produces friction and heat. Some designs resist the wheel driving the worm backward, but self-locking is conditional, not guaranteed.
KHK · Worm gearsRack & pinion: straight-line motion.
A round pinion engages a straight toothed rack. Rotation can move the rack, or move the pinion assembly along a fixed rack. A direct way to convert between rotation and linear travel, including long travel distances. Backlash can create lost motion when reversing. Accurate mounting and alignment are important.
KHK · Gear racksPlanetary: compact coaxial drive.
Planet gears sit between a sun and an internal ring, linked by a carrier. In the common reduction shown, the ring stays fixed, the sun drives, and the carrier turns the output. A compact arrangement with input and output on the same axis, sharing load across several planets. More parts and constraints. The ratio depends on which member is fixed, driven, and used as output. The simple two-gear formula alone cannot describe it.
Neugart · Planetary gearboxesWorm-drive self-locking is conditional.
Whether the wheel can drive the worm depends on lead angle and friction conditions. It is not guaranteed, and this page does not model load holding.
KHK · §4.6(4), Self-Locking of Worm Gear PairA 1:1 miter pair changes shaft direction without changing speed magnitude; ideal spur gears produce no axial thrust.
KHK distinguishes spur, helical, bevel, and miter designs and their shaft arrangements.
KHK · Types of GearsA robot’s wheel: more turning torque at the wheel.
A reduction lets the output turn more slowly while increasing its ideal torque. Real robot gearboxes may combine several stages. The source provides application context; our 18→36 tooth-count preset is illustrative, not a product replica.
Pololu · Balboa external gearingA hand pulling winch: more turns of the handle, more drum torque.
The handle turns a smaller gear, which drives a larger one connected to the drum. The trade gives more output torque for a slower drum. The source provides application context; our 18→60 tooth-count preset is illustrative, not a product replica.
Dutton-Lainson · Hand winchA hand-cranked spinner: a faster output from a slower hand.
Turn the larger gear to drive a smaller one. The pointer gains speed while giving up ideal turning torque. The source provides application context; our 36→18 tooth-count preset is illustrative, not a product replica.
LEGO Education · Gear activitiesSprockets engage a chain, rather than meshing directly with another gear.
A bicycle chain drive should not be pictured as the directly meshing spur pair used in this experiment.
U.S. Tsubaki · SprocketsWhat the model assumes
- Two unshifted external spur gears, common module 2 mm, pressure angle 20°, full-depth teeth. Allowed tooth counts: 18, 24, 30, 36, 48, 60.
- Involute flanks, with a simplified unfilleted root and no manufacturing tolerances. This is a teaching schematic, not a manufacturing drawing.
- The minimum tooth count is conservative for this standard geometry. It is not a universal minimum for every gear design.
- Rigid parts, ideal zero backlash, no friction, deformation, or inertia. Input RPM is prescribed. The torque comparison assumes a fixed illustrative 1 N·m input in steady, lossless transmission.
- Changing tooth count rebuilds and repositions the pair, restarting the rotation. Units change labels only. This model does not predict motor startup, load response, efficiency, or tooth strength.
- Web openings, shafts, hubs, and the support plate illustrate an assembly. They are not validated for material strength or manufacturing. Reference circles and the line of action explain geometry, not measured forces.
- One-turn mode resets the input marker, advances exactly one revolution, and stops at the corresponding output angle. Continuous motion and manual rotation use the same tooth-count relationship. Counters show net angular displacement from the reset, in turns.
- The six family diagrams are static arrangement schematics. They do not simulate tooth contact, force, or efficiency. The planetary diagram shows a fixed ring, sun input, and carrier output; it is not a separate validated planetary simulation.
- Application presets use the main external spur model at 15 RPM to make the motion easy to follow. They do not reproduce the cited products or model a motor, load, traction, winch capacity, or holding mechanism.
- Pitch circles roll together in this ideal model. Actual tooth surfaces generally slide relative to each other except at the pitch point.
Independent subject review is pending.
Read the sources and model assumptions