Gear - Wikiparency

# Gear A **gear** is a rotating [[Machine|machine]] element with cut teeth that mesh with the teeth of another toothed part to transmit [[Torque|torque]] and motion. Because the teeth engage instead of rubbing, a pair of gears turns in a fixed ratio set by their tooth counts, and that ratio holds exactly, turn after turn, in a way no belt or friction wheel can promise. A gear meshed with a part of different tooth count changes the speed and the torque together: what it multiplies in one it divides in the other, leaving the [[Power_(physics)|power]] that passes through almost unchanged. The relations that matter are short. For two meshing gears with N₁ and N₂ teeth turning at n₁ and n₂, `n₁N₁ = n₂N₂`, so the speed ratio is the inverse of the tooth ratio, while the torque ratio is the tooth ratio itself. The size of the wheel that carries those teeth follows from the module m, the tooth size: the pitch diameter is `d = mN`, and two meshing gears sit at a centre distance `C = m(N₁ + N₂)/2`. These four relations decide most of what a gearbox looks like before any strength calculation begins, because a large ratio in a single mesh demands a large wheel. The rest of the subject is about making the ratio smooth and making it last. A tooth profile is chosen so that the ratio holds not just from tooth to tooth but continuously through each engagement — the [[Involute|involute]] profile does this and tolerates errors in centre distance while doing it. Teeth are cut straight, helical or worm-like according to how much noise, thrust and ratio the drive must take; they are sized by standard modules or diametral pitches so that parts interchange; and they are rated against [[Fatigue_(material)|fatigue]], pitting and scuffing by standards that the industry shares. The framework microsim *Gears: teeth trade speed for torque, and power passes through untouched* draws a meshing pair to scale with true involute teeth: the reader changes the tooth counts and the module and watches the speed fall, the torque rise, the two power bars stay level, and the output gear grow across the frame until a single-mesh reduction stops being buildable. ## History The earliest surviving toothed wheels are ancient. Iron gears from the Warring States period, preserved at the [[Luoyang|Luoyang]] Museum, place working gearing in China in the fourth century BC, and in Europe [[Aristotle|Aristotle]] mentions toothed wheel drives in windlasses around 330 BC, noting that the sense of rotation reverses at each mesh.[^history] The most sophisticated surviving ancient example is the [[Antikythera_mechanism|Antikythera mechanism]], a Greek geared calculator recovered from a first-century BC shipwreck; surface imaging and computed tomography published in 2006 resolved many of its inscriptions and gear trains, showing a device that modelled the motions of the Sun and Moon with more than thirty bronze wheels.[^freeth2006] For most of the following two millennia gearing was the business of mills, clocks and hoists. Large wheels were built of wood with inserted pegs, an arrangement that gave the English word *cog* its mechanical sense and made a worn tooth a replaceable part rather than a scrapped wheel. Precision came from [[Clock|clockmaking]], where a train of small wheels had to divide a falling weight's motion into hours and minutes without accumulating error, and where the [[Escapement|escapement]] made the demand for repeatable tooth geometry unavoidable. Two shifts made gearing modern. The first was mathematical: the recognition, worked out in the seventeenth and eighteenth centuries, that a constant speed ratio requires a particular conjugate tooth shape, and that the [[Cycloid_gear|cycloidal]] and involute curves both satisfy it. The second was industrial: machine tools that could cut those curves accurately in metal, above all the hobbing machines of the late nineteenth century, which generate a tooth profile by rolling a cutter against a blank rather than copying a template. From that point gear ratios could be specified rather than fitted, and a gear became a catalogue item. ## Etymology The word *gear* comes from [[Old_Norse|Old Norse]] *gørvi*, meaning apparel or equipment, related to a verb for making, building or setting in order; the general sense of equipment survives in phrases such as fishing gear. The specific mechanical sense of a toothed wheel in machinery is recorded from the 1520s, and the sense of a set of ratios in a vehicle [[Transmission_(mechanical_device)|transmission]] is later still.[^etymonline] *Cog*, the older English word for a single tooth and then for a toothed wheel, keeps the memory of the wooden pegs that were driven into the rim of a mill wheel. ## Materials Material choice follows the load, the speed and the noise the drive can tolerate. The [[Antikythera_mechanism|Antikythera mechanism]]'s wheels are [[Bronze|bronze]], the earliest surviving Chinese gears are iron, and both metals, with [[Cast_iron|cast iron]], carried gearing through the clock and mill era; large mill wheels were commonly wood, with hardwood cogs set into a cast rim. Today the default for a loaded mesh is steel, usually a low-alloy carburising grade whose teeth are [[Case-hardening|case-hardened]] to resist pitting while the core stays tough enough to absorb shock — a deliberately graded material, hard where the contact is and ductile where the bending stress runs. Where loads are light, other properties win. Cast iron damps vibration and machines cheaply for slow, heavy wheels. Bronze is used for worm wheels because it slides well against a hardened steel worm, which is the one common mesh with large sliding rather than rolling contact. Polymers — principally [[Nylon|nylon]] and [[Polyoxymethylene|acetal]] — run quietly, need no lubricant and cost little in volume, at the price of low strength, [[Creep_(deformation)|creep]] under sustained load and a size that changes with temperature and moisture. The trade is visible in any domestic appliance, where a nylon gear meshes with a steel pinion and is expected to be the part that fails. ## Manufacture Gears are made either by cutting teeth into a blank or by forming them to shape. Cutting remains the route for accuracy: [[Hobbing|hobbing]], in which a rotating worm-shaped cutter is rolled against the blank so that the tooth flank is generated as an envelope, is the most common method, with gear shaping for internal gears and shoulders that a hob cannot reach, and [[Grinding_(abrasive_cutting)|grinding]] or honing afterwards where the drive must be quiet or precise. Forming routes include die, sand and investment [[Die_casting|casting]], [[Injection_moulding|injection moulding]], [[Powder_metallurgy|powder metallurgy]] — which needs a sintering step after the part leaves the mould — and blanking from sheet. One trade survey estimated that as of 2014 about 80 percent of gearing produced worldwide came from net-shape moulding rather than cutting, the great majority of it small and lightly loaded.[^eberle] Heat treatment is part of manufacture, not an afterthought. Carburising and quenching raise surface hardness where the teeth touch; large gears prone to distortion are quenched in a press that holds them flat while they transform. Because hardening moves the metal, precision gears are ground after treatment, which is why a quiet automotive gear costs several times what its blank and cutting alone would suggest. [[3D_printing|Additive manufacture]] appears mainly in prototypes and very small runs, where accuracy and strength matter less than the week saved. ## Comparison with other drive mechanisms The alternatives to a gear train are chains on sprockets, belts on pulleys, friction wheels and hydraulic couplings, and the comparison turns on one property: a gear mesh does not slip. Tooth engagement ties the two shafts together so that rotation tracks exactly, limited only by [[Backlash_(engineering)|backlash]] and manufacturing error, which is why gearing is chosen wherever position rather than mere power must be transmitted — in clocks, instruments, machine tools and [[Robotics|robot]] joints. A single mesh is also compact, needs only two moving parts, and is the most efficient practical way to connect shafts that are not parallel. The costs are equally clear. Gears are expensive to make to the tolerances they need, most gear drives require lubrication and a housing to retain it, and a meshing pair usually has more mass and [[Moment_of_inertia|rotational inertia]] than the [[Belt_(mechanical)|belt]] and pulleys it replaces. The centre distance is fixed once the gears are cut, so a gear drive cannot absorb the variable shaft spacing that a [[Roller_chain|chain]] or belt takes in its stride. And slip is sometimes the point: a belt that slips under a shock load, or a friction drive that gives way, protects the machine behind it in a way that a gear train will not — a gear tooth simply breaks. ## Ideal gear model For analysis a gear is idealised as a perfectly [[Rigid_body|rigid body]] turning about a fixed axis, so every point moves on a circle about that axis at the same [[Angular_velocity|angular speed]] ω(t). What matters of its surface is the action surface, the part that can bear against its partner: in a gear of N teeth it repeats with N-fold rotational symmetry, one patch per tooth face, and a drive that carries torque both ways uses two such sets, one per flank. Meshing is then a pair of imaginary pitch circles rolling on each other without slipping, and the teeth exist only to make that rolling positive. Everything the model gives follows from the rolling condition. Equal pitch-line speed at the mesh gives `n₁N₁ = n₂N₂`, so the output turns slower by exactly the tooth ratio, and reverses sense at every external mesh. A lossless mesh passes the same power at each shaft, `P = Tω`, so the torque grows by the same factor as the speed falls: `T₂ = T₁N₂/N₁`. A wheel of N teeth at module m has pitch diameter `d = mN`, and the two shafts stand at `C = m(N₁ + N₂)/2`. The worked example the sim opens on comes from the design text behind it: a 20-tooth pinion at 1,800 rpm carrying 10 N·m drives a 600-tooth wheel, so the output turns at `1800 × 20/600 = 60` rpm with `10 × 600/20 = 300` N·m, and both shafts pass `10 × 1800 × 2π/60 = 1,885` W.[^jensen224] At module 2 those counts mean a 40 mm pinion, a 1,200 mm wheel and a 620 mm centre distance — a wheel more than a metre across for one 1:30 reduction, the reason such ratios are split between stages. The sim's mesh is lossless and rigid by construction — no friction, no tooth deflection, no backlash — so its two power bars are exactly equal, where a real spur mesh gives back a percent or two less than it takes.[^gearspec] *Try: raise the output teeth to 600 and watch the speed fall to 60 rpm, the torque climb to 300 N·m, the two power bars stay level at 1,885 W, and the output gear grow past the frame — the 1.2 m wheel a single-mesh 1:30 reduction really needs; then halve the module to 1 and watch every diameter and the centre distance halve while all three ratios stay put.* ## Relative axis position The first way to classify a meshing pair is by how the two axes lie. Parallel axes are the common case: the gears contact between the two shafts and turn in opposite senses, unless one is nested inside the other as an internal gear, in which case they turn the same way. A parallel pair can be analysed slice by slice, as a stack of infinitesimally thin flat gears, which is why the two-dimensional picture of pitch circles is enough for it. Axes that cross at an angle call for [[Bevel_gear|bevel gears]], each shaped as a frustum of a cone whose apex sits at the point where the axes meet; equal-toothed bevels at 90° are called mitre gears. Axes that are skew — neither parallel nor intersecting, the arrangement of a [[Worm_drive|worm]] drive or a car's hypoid final drive — cannot be matched by cones at all, and their ideal pitch surfaces are portions of a hyperboloid of revolution. Hypoid teeth slide along each other as they roll, which makes their contact gradual and quiet but demands the heavily additived [[Lubrication|gear oils]] sold for that duty. ## Tooth orientation A gear is external if its teeth point away from its axis and internal if they point inward, and in a matching pair only the larger wheel can be internal. The internal arrangement is compact — the two axes sit close together, the pair turns in the same sense, and the contact is between a convex and a concave flank, which spreads the load — and it is what makes [[Epicyclic_gearing|epicyclic]] trains possible. A crown or contrate gear is the limiting case in which the teeth project at right angles to the wheel's plane, meshing with a pinion whose axis lies along the crown's face; crown gears are found in hand drills and in mechanical [[Clock|clocks]], where one may mesh directly with the [[Escapement|escapement]]. ## Tooth cut direction Teeth run across the width of a gear in one of a few directions. Straight-cut spur teeth are parallel to the axis: cheapest to make and to inspect, with no axial thrust, but each tooth takes up its share of the load along its whole length at once, which makes spur gears noisy at speed. Helical teeth are cut at an angle, so contact begins at one end of a tooth and sweeps across it; more than one tooth pair is always in contact, load transfer is gradual, and the drive is markedly quieter, at the cost of an axial thrust the bearings must take. Double-helical or [[Herringbone_gear|herringbone]] teeth put two opposite helices on one wheel so that the thrusts cancel, which is why they appear on large, heavily loaded reduction gearing. A worm is a helical gear taken to its limit: a helix angle near 90° and a body long enough that at least one tooth wraps the axis completely, so that it looks and acts like a screw meshing with a wheel. The arrangement is compact and gives in a single stage ratios that a helical pair cannot approach. It is also the one common gear mesh dominated by sliding rather than rolling, so its efficiency is low and its friction high — and that is exactly why a worm drive is often self-locking, holding its load when the input is released, the same friction-versus-geometry trade that governs every [[Simple_machine|simple machine]] with sliding contact. ## Tooth profile The tooth profile is the shape of a tooth's cross-section in a plane cut perpendicular to the pitch surface, and it decides whether the speed ratio is really constant. Hand-cut artisanal teeth, such as those of the [[Antikythera_mechanism|Antikythera mechanism]], were often simple triangles, which mesh but do not hold a constant ratio; a lantern or cage gear, whose teeth are round pins held between two discs, was the robust mill and clock answer for centuries and is still used where dirt would ruin a cut flank. Modern profiles are mathematical: the requirement of conjugate action — that the common normal at the contact point always pass through the pitch point — is what forces a particular curve. Two curves satisfy it for parallel and crossed axes. The [[Cycloid_gear|cycloidal]] profile, still standard in horology, gives low friction near the pitch point and tolerates small tooth counts. The involute of a circle dominates everywhere else, because it has a property the cycloid lacks: the [[Line_of_action|line of action]] is straight and fixed, inclined at the [[Pressure_angle|pressure angle]] φ, and the speed ratio is unchanged if the centre distance is slightly wrong. A drive can therefore be assembled with ordinary tolerances and still hold its ratio, and one cutter generates every tooth count at a given module. For skew axes with curved tooth traces the corresponding answer is one of the [[Spiral_bevel_gear|spiral bevel]] families. The sim draws true involutes rather than schematic teeth: each flank is generated from the base circle `r_b = r_p cos φ`, with an addendum of one module above the pitch circle, a dedendum of 1.25 modules below it, and a tooth thickness of `πm/2` at the pitch circle.[^gearspec] For the 20-tooth, module 2 pinion above, that means a 40 mm pitch diameter, an 18.79 mm base circle at the standard 20° pressure angle, a 44 mm tip diameter, a 35 mm root diameter, a circular pitch of 6.28 mm and a tooth 3.14 mm thick at the pitch line. Those numbers are what the picture is drawn from, which is why the teeth stay the same size as the wheel grows and the tooth count, not the tooth, is what changes. ## Special gear trains Two arrangements are special enough to have their own names. A [[Rack_and_pinion|rack]] is a gear of infinite radius — a toothed bar — and a pinion running along it converts rotation into translation at `v = ω r_p`, the basis of steering boxes, machine-tool traverses and the rack railway. An [[Epicyclic_gearing|epicyclic]] or planetary train is the case in which one or more gear axes themselves move: planets ride on a carrier between a central sun and an internal ring, and which member is held fixed decides the ratio. With the ring stationary and the sun driven, the carrier turns at `1 + N_ring/N_sun` times slower than the sun — four to one for a 24-tooth sun in a 72-tooth ring — while the load is shared between several planets, so a planetary stage carries more torque for its size than any single mesh and keeps its input and output on one axis. The [[Sun_and_planet_gear|sun and planet gear]] is the historically important variant, used by [[James_Watt|James Watt]] to convert the reciprocating motion of a [[Steam_engine|steam engine]] into rotation; it doubles the output shaft's speed relative to the beam's stroke, a side effect that suited the mills it drove. Compounding is the other standard move: two or more meshes in series, with the intermediate wheels on a common shaft, multiply their ratios. It is what keeps a large reduction small. Two stages of 20:110 teeth give 30.25 overall at module 2 with a largest wheel of 220 mm, a fifth of the diameter of the single 600-tooth wheel that produces the same 1:30 in one mesh, and the minimum-sized train for a given ratio comes near the square root of it in each stage.[^jensen224] ## Non-circular gears A gear need not be round. [[Non-circular_gear|Non-circular gears]] — elliptical wheels are the common form — mesh with a partner shaped to complement them, so that the speed ratio varies continuously through each revolution while the teeth still engage positively. The aim is no longer smooth, constant transmission but a wanted variation: a quick return on one half of the turn, a controlled oscillation of the output shaft, or a ratio that sweeps through a range. They appear in textile machinery, printing presses, flow meters, [[Continuously_variable_transmission|continuously variable transmissions]] and potentiometer drives, and they are laid out by the same conjugate-action condition as a round gear, applied to a rolling pitch curve that is not a circle. ## Non-rigid gears Some gearing abandons the rigid-body picture entirely. A harmonic or [[Strain_wave_gearing|strain wave]] drive transmits torque through a thin, deliberately flexible cup — the flexspline — which an elliptical wave generator deforms so that its external teeth engage an internal circular spline at two opposite places. Because only the small difference in tooth count between the two splines advances per revolution of the wave generator, one stage gives reductions of the order of a hundred to one, with essentially zero backlash and coaxial shafts. Those properties make it the standard reducer in the joints of [[Robotics|industrial robots]], where lost motion at the joint is lost accuracy at the tool. A [[Magnetic_gear|magnetic gear]] goes further and removes contact: the "teeth" are magnetic poles, and torque crosses an air gap through the field. Nothing touches, so nothing wears and no lubricant is needed, and the drive slips harmlessly instead of breaking when overloaded — an inherent torque limit that a toothed mesh can only get from an added clutch. The costs are a lower torque density than steel teeth of the same size, sensitivity to temperature through the magnets, and magnet cost. ## Nomenclature Gear vocabulary is standardised, and most of it names a circle. The pitch circle is the imaginary circle that rolls on its partner's; its diameter d is the pitch diameter, and the pitch point is where the two touch. The addendum is the tooth height above the pitch circle and the dedendum its depth below, their sum being the working depth plus clearance; the face width is the axial length of the teeth. Circular pitch is the arc distance from one tooth to the next measured on the pitch circle, `p = πm`, and tooth thickness is normally half of it. The [[Pressure_angle|pressure angle]] φ — 20° in most modern practice, 14.5° in older work — fixes the [[Line_of_action|line of action]] along which the teeth push, and hence the base circle `r_b = r_p cos φ` from which an involute is generated.[^gearspec] One term is used in two opposite senses and is worth stating carefully. *Gear ratio* is applied both to the tooth ratio N₂/N₁ and to its reciprocal, the speed ratio n₂/n₁, so a ratio quoted without a direction is ambiguous: the design text behind this article's sim defines it as the tooth ratio, while the sim's own equation line states the speed form.[^jensen224][^gearspec] Three further terms carry most of the design argument. The contact ratio is the average number of tooth pairs in contact at once; below 1.2 or so a spur mesh becomes rough, and raising it is much of the reason for helical teeth. The helix angle sets how far a helical tooth advances across the face, and with it both the overlap and the axial thrust. [[Backlash_(engineering)|Backlash]] is the clearance between mating flanks. A full vocabulary, with the symbols, is maintained by the American Gear Manufacturers Association together with ANSI in its nomenclature standard,[^agmanom] and the terms are collected on the companion page [[List_of_gear_nomenclature|list of gear nomenclature]]. ## Backlash Backlash is the lost motion that appears when a gear pair reverses direction: a gap always exists between the back of a driving tooth and the tooth behind it on the driven wheel, and that gap must be closed before force is transmitted the other way. The word names both the phenomenon and the size of the gap, which is normally quoted as an arc length at the pitch circle. It is deliberate. A mesh built to exactly zero clearance would bind as soon as the housing warmed, the shafts deflected or a film of oil entered, so backlash is specified, produced by thinning the teeth slightly, and expected to grow as the flanks [[Wear|wear]]. Its cost is position. A 0.05 mm backlash on a 40 mm pitch-diameter pinion is 0.05/20 = 2.5 milliradians of lost motion, about 0.14° at that shaft, which appears at the output multiplied or divided by whatever ratios lie between — trivial in a conveyor drive, fatal in a machine-tool axis or a [[Robotics|robot]] wrist. Where it matters, it is engineered out: split "scissor" gears whose two halves are sprung apart to fill the gap, preloaded duplex worms, adjustable centre distances, or a control system that always approaches a position from the same side. The other route is to avoid the mesh, which is one of the reasons [[Strain_wave_gearing|strain wave]] and preloaded ball-screw drives are used in precision machinery. ## Standard pitches and the module system Gears can be cut to any pitch, but tooth sizes are standardised so that cutters, blanks and stock gears interchange. Inch practice uses diametral pitch, the number of teeth per inch of pitch diameter, `P = N/d`, with preferred coarse values such as 2, 4, 6, 8, 10, 12 and 16, and fine pitches above 20; a larger number means a smaller tooth. Metric practice uses the module, `m = d/N` in millimetres — a direct length, so a larger number means a larger tooth — in a preferred series such as 1, 1.25, 1.5, 2, 2.5, 3, 4, 5 mm. The two measures are reciprocal through the inch: `P = 25.4/m`, so the module 2 of the sim's default is a diametral pitch of 12.7, which is why a metric gear and an inch gear of similar size do not mesh. Standard tooth sizes are what make the rest of the arithmetic tractable. Once m and the tooth counts are chosen, the pitch diameters `d = mN` and the centre distance `C = m(N₁ + N₂)/2` follow with no further freedom, the addendum and dedendum are the conventional multiples of m, and a single [[Hobbing|hob]] cuts every tooth count at that module — the interchangeability that lets a [[Gear_train|gear train]] be assembled from catalogue parts. It also means that the module is the one control that changes the drawing without changing the drive: in the sim, halving m from 2 to 1 halves both diameters and the centre distance while the speed, torque and power ratios stay exactly as they were — the ratio lives in the tooth counts, the size lives in the module.[^gearspec] ## Gear failure mechanism Gear teeth fail in a handful of well-characterised ways, often several at once, and which one dominates depends on load, speed and lubrication. Surface mechanisms come first: [[Wear|wear]] from abrasion or adhesion; scuffing, where the oil film breaks down under load and speed and the flanks weld and tear; pitting and micro-pitting, the contact-fatigue mechanisms in which repeated Hertzian pressure at the rolling contact spalls metal out of the flank. Bulk mechanisms follow: tooth flank fracture, which starts below the case, and tooth root fatigue fracture, where a bending crack grows at the root fillet until the tooth breaks off. The driving phenomena are [[Friction|friction]], [[Contact_mechanics|contact]] pressure with its mix of rolling and sliding, bending [[Fatigue_(material)|fatigue]], and insufficient [[Lubrication|lubrication]]. Because the mechanisms compete, rating a gear means checking more than one limit. The international standard ISO 6336 and the American ANSI/AGMA 2001 both compute a load capacity for surface durability and another for tooth root strength, with factors for application, dynamic load, load distribution across the face and size, so that a design is limited by whichever governs.[^iso6336][^agma2001] In service, the same mechanisms announce themselves before they finish: pitting and cracking change the [[Vibration|vibration]] signature of a gearbox at tooth-mesh frequency and its sidebands, which is what [[Condition_monitoring|condition monitoring]] listens for on wind turbines and industrial drives. ## Gear model in modern physics Physics has borrowed the gear as a model more than once. In the nineteenth century [[James_Clerk_Maxwell|James Clerk Maxwell]] built a mechanical picture of the electromagnetic field in which magnetic field lines were rotating tubes of fluid, and, needing neighbouring tubes to turn the same way, inserted small "idle wheels" between them — the idler of a [[Gear_train|gear train]] — identifying their rotation with electric current. The model was scaffolding, not a claim about the ether's contents, and Maxwell said as much; it nonetheless carried him to the field equations.[^maxwell1861] The image recurs at much smaller scale. Groups of interlocking rotors are used as models of nanomechanical devices and of ring molecules whose rotations are coupled, where the question is whether rotation of one unit drives its neighbour with a fixed phase relation, as a gear train would, or slips. The vocabulary — gear, mesh, slip, ratio — transfers because the underlying constraint is the same: rigid coupling between rotations, imposed by geometry rather than by force. ## Gear mechanism in natural world Gearing was long thought to be an exclusively human invention. It is not. The hind legs of [[Planthopper|planthopper]] nymphs carry intermeshing gear teeth on the inner faces of their trochanters, and in 2013 Malcolm Burrows and Gregory Sutton filmed the jump of *[[Issus_coleoptratus|Issus coleoptratus]]* nymphs at high speed and showed what the gears are for: each leg carries a strip of teeth about 400 micrometres long on a pitch radius of about 200 micrometres, with ten to twelve teeth whose roots are filleted exactly as an engineer would fillet them against shear.[^burrows2013] The function is synchronisation rather than ratio change. Both legs must extend together, and a nervous system cannot coordinate them finely enough; the interlocking teeth mechanically lock the two legs' rotations so that they start within about 30 microseconds of each other, preventing the yaw that would spin the insect off course. The teeth engage only when the animal cocks its legs to jump, and the whole mechanism is lost at the final moult to adulthood, where friction between the leg surfaces takes over — the suggested reason being that a broken tooth cannot be replaced once an insect has stopped moulting. It remains the clearest known case of a functioning mechanical gear that evolved rather than being designed.[^burrows2013] ## See also - [[Gear_train]] - [[Epicyclic_gearing]] - [[Rack_and_pinion]] - [[Involute_gear]] - [[Strain_wave_gearing]] - [[Worm_drive]] - [[Bevel_gear]] - [[Backlash_(engineering)]] - [[List_of_gear_nomenclature]] - [[Differential_(mechanical_device)]] - [[Sprocket]] - [[Kinematic_chain]] - [[Simple_machine]] ## References [^jensen224]: Jensen, N. *Introduction to Mechanical Design and Manufacturing*. Portal Book 109, pp. 224–226: gear types, gear terms, the ratio relations, and the worked reduction from 1,800 rpm and 10 N·m to 60 rpm and 300 N·m with 20 and 600 teeth at module 2 (centre distance 620 mm, 1,885 W); p. 215 for the mechanical advantage of gears. The text defines *gear ratio* as the tooth ratio N₂/N₁ on p. 224, where the sim's equation line uses the reciprocal speed ratio n₂/n₁. [^gearspec]: Engineering portal sim spec `specs/sims/Gear.json`, framework module `design.gear`: `mesh` returns the ratio from `n₁N₁ = n₂N₂` with `T₂ = T₁N₂/N₁` at η = 1, `d = mN`, `C = m(N₁ + N₂)/2`, `P = Tω` and the reversed sense; `outline` draws true involutes generated from the base circle `r_b = r_p cos φ`, addendum m, dedendum 1.25 m, tooth thickness `πm/2` at the pitch circle. The mesh is modelled as lossless and rigid — no friction, no tooth deflection, no backlash — so the sim's two power bars are exactly equal by construction; the drawn geometry is to scale rather than schematic. [^eberle]: Eberle, Fred (August 2014). "Materials Matter." *Gear Solutions*, p. 22 (trade press). http://www.gearsolutions.com/article/detail/6458/materials-matter-fred-eberle [^etymonline]: *Online Etymology Dictionary*, entry "gear" (Old Norse *gørvi* 'apparel, gear'; the sense 'toothed wheel in machinery' attested from the 1520s). https://www.etymonline.com/word/gear [^freeth2006]: Freeth, T.; Bitsakis, Y.; Moussas, X.; et al. (November 30, 2006). "Decoding the ancient Greek astronomical calculator known as the Antikythera Mechanism." *Nature* 444 (7119): 587–591. https://doi.org/10.1038/nature05357 [^burrows2013]: Burrows, Malcolm; Sutton, Gregory (September 13, 2013). "Interacting gears synchronize propulsive leg movements in a jumping insect." *Science* 341 (6151): 1254–1256. https://doi.org/10.1126/science.1240284 [^iso6336]: International Organization for Standardization (2019). ISO 6336-1:2019, *Calculation of load capacity of spur and helical gears — Part 1: Basic principles, introduction and general influence factors*. https://www.iso.org/standard/36327.html [^agma2001]: American Gear Manufacturers Association. ANSI/AGMA 2001, *Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth* (surface durability and bending strength ratings; edition letter not pinned for this run). [^agmanom]: American Gear Manufacturers Association and American National Standards Institute. *Gear Nomenclature, Definitions of Terms with Symbols* (standard number and edition not pinned for this run). [^maxwell1861]: Maxwell, James Clerk (1861–1862). "On physical lines of force." *Philosophical Magazine*, series 4, in four parts. (Volume and page numbers as recalled and not pinned for this run; the idle-wheel construction and Maxwell's own description of the model as illustrative are standard in histories of electromagnetism.) [^history]: *Citation needed.* The dates in this section's first paragraph — iron gearing of the Warring States period held at the Luoyang Museum, and Aristotle's mention of toothed wheel drives around 330 BC — are standard in gear histories but are not supported by any Portal Book carried in this run, whose engineering sources begin with the modern mesh. A museum catalogue entry for the Luoyang gears and a modern edition of the pseudo-Aristotelian *Mechanical Problems* are the records that would settle them.  **Microsim — three.js (Wikitube framework):** *Gear* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/engineering/Gear.html" data-title="Gear"></div> *Built from `MICROSIM_GUIDE/specs/sims/Gear.json`; part of the [[PORTAL_Engineering|Engineering portal]] spine (section sims and See-also variants).*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Gear) : [Wikitube](https://en.wikitube.io/wiki/Gear) - skeleton pinned to revision 1375068669 (2026-09-18).