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 TechnologyA turn for the better: sources & model
An ideal kinematic model of two external spur gears, supported by practical application stories and a comparison of six gear designs and arrangements. The main experiment prescribes input rotation and calculates output rotation.
Scientific review · independent subject review pending
The source and model records are available for inspection. No external scientific reviewer has signed off yet.
gears-content-3 · spur-gears-1 · configuration format 1What supports the explanation?
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 this 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.
What has been checked
Reference ratios, direction, ideal power, dimensions, permitted inputs, polygon engagement, one-turn stopping in both directions, and reference contact geometry are checked numerically. Independent mechanical subject review remains pending.
Each source supports the associated claim. Sources do not certify this implementation or its visuals.
From gears to a chain drive
bicycle-chain-1 · setup format 1
The bicycle applies a rotation ratio through an uncrossed chain. Both sprockets turn in the same direction. Its ratio is expressed as wheel turns per pedal turn, the reciprocal of the speed-reduction convention used for the external spur pair.
The bicycle ratio is front teeth ÷ rear teeth.
One front revolution moves one chain pitch per front tooth; count how many revolutions that produces at the rear sprocket.
Exploratorium · Count your teethCadence, tooth ratio, and wheel circumference determine engaged rolling speed.
Activity 3.3 and its worksheet connect wheel circumference, pedal revolutions per minute, and travel. Our wheel radius of 0.35 m is an explicit illustrative assumption.
Canada Science and Technology Museum · Cycle-ology, Activity 3.3The wheel may overrun the rear sprocket.
A freehub cannot engage while the wheel hub rotates faster than the cassette. Our model uses ideal immediate engagement, omitting the finite angle and real friction.
DT Swiss · Engagement Angle, Additional remarksReal roller chains have discrete, polygonal motion.
The original mechanism authors analyze exact and approximate chain kinematics. Our smooth circular path preserves mean tooth-count ratio and omits small variations.
Fuglede & Thomsen · Kinematics of roller chain drives (2016)Bicycle chain teeth engage rollers; these are not involute spur gears.
Modern bicycle chains use 12.7 mm (½ inch) pitch. Pitch alone does not establish component compatibility.
Park Tool · Chain CompatibilityThe tooth sizes are examples found across actual product families.
32T appears in SRAM Eagle options; 40T and 48T appear in eTap AXS configurations. This does not make the model a compatible purchasable assembly.
SRAM · eTap AXS chainring sizes32T is a listed Eagle chainring option.
The manufacturer lists this size among its front-chainring options.
SRAM · Eagle chainring sizes16T, 24T, and 32T are actual rear sprocket sizes.
The XG-1275 specifications include these tooth counts within cassette sequences. The visualization shows one selected sprocket, not a complete cassette.
SRAM · XG-1275, cog sizesReal rider torque varies, unlike a simple ideal steady comparison.
The cited efficiency test discusses pedal-torque variation and explicit test conditions. We do not use it to claim a universal drivetrain efficiency.
Rohloff · Efficiency MeasurementImplemented assumptions and checks
- Front teeth 32, 40, or 48; rear 16, 24, or 32. A single chain line with parallel sprocket axes and a schematic freehub. Real cassette geometry, derailleur routing, shifting, chain-length adjustment, and take-up are omitted. Replacing either sprocket resets the comparison.
- Rolling radius 0.35 m; no slip. Both wheels roll at the same angular speed because their radii match. The chain drives only the rear wheel. Moving ground represents forward travel.
- Mean chain-path radius Np/(2π), with p = 12.7 mm. The real pin-center polygon has radius p/[2 sin(π/N)]. The displayed links, tooth profiles, frame, and other parts are schematic; this is not a manufacturing or fit model.
- Pedaling cadence is prescribed from 0 to 120 RPM, with a 15 RPM/s ramp. When driven sprocket speed catches wheel speed, the ideal freehub engages. Otherwise the wheel overruns. No driving torque is transmitted while overrunning.
- Ideal coasting has no resistance and preserves wheel speed. “Slow to a stop” prescribes 0.8 m/s² deceleration with the pedals stopped. There is no rider force, inertia, road grade, tire traction, air drag, efficiency, or actual braking calculation.
- The torque factor is an ideal steady, engaged comparison. It does not describe the force causing the prescribed acceleration. Fixed 1/120 s steps; real-time or quarter-speed playback. Pausing stops simulated time.
- Setup links store tooth counts, target cadence, view, and display units. They reopen paused from rest and do not replay a ride. Bicycle export generates a 20-second side-view diagram from the current model state and action at real-time speed, independently of slow playback. It uses the same fixed-step model.
- Numerical checks cover reference speeds, ratio and ideal power, independent wheel overrun, catch-up, nonnegative braking, rolling distance, and a continuous tangent chain path across all supported pairs. These checks do not replace mechanical or learner review.
Try it at home: evidence and limits
spool-belt-1 · A friction-belt analogy for linked rotation. NREL’s Activity Four uses unequal marked spools and a rubber band. Its later transmission hints discuss slipping and stretching.
Brytalearn’s adaptation replaces the source’s nailed mounting with blunt axles held parallel by a helper. No cutting or powered rotation is proposed. This mounting adaptation has not yet been physically trialed by Brytalearn. Independent subject review remains pending.
The band’s contact diameter matters, not the spool’s outer flange. For an approximately non-slipping belt, turns ratio is approximately the inverse contact-diameter ratio. The on-screen diagram uses a 2:1 diameter ratio with no slip; real household supplies can yield different and inconsistent counts.
The activity does not reproduce chain tooth engagement, freehub behavior, or load forces. Observation notes are local to the page; the learner can download them with the steps. No supply purchases or account are required.
NREL · Activity Four (page 8) & transmission hints (page 14)