Un engranaje es una parte de una máquina circular giratoria que tiene dientes cortados o, en el caso de una rueda dentada o rueda dentada , dientes insertados (llamados engranajes ), que engranan con otra parte dentada para transmitir el par . Un engranaje también puede conocerse informalmente como un engranaje . Los dispositivos con engranajes pueden cambiar la velocidad, el par y la dirección de una fuente de energía . Los engranajes de diferentes tamaños producen un cambio en el par, creando una ventaja mecánica , a través de su relación de transmisión , y por lo tanto pueden considerarse una máquina simple . Las velocidades de rotación, y los pares, de dos engranajes difieren en proporción a sus diámetros. Los dientes de los dos engranajes tienen la misma forma. [1]
Dos o más engranajes engranados, que trabajan en secuencia, se denominan tren de engranajes o transmisión . Los engranajes de una transmisión son análogos a las ruedas de un sistema de polea de correa cruzada . Una ventaja de los engranajes es que los dientes de un engranaje evitan el deslizamiento. En transmisiones con múltiples relaciones de transmisión, como bicicletas, motocicletas y automóviles, el término "marcha" (por ejemplo, "primera marcha") se refiere a una relación de transmisión en lugar de una marcha física real. El término describe dispositivos similares, incluso cuando la relación de transmisión es continua en lugar de discreta, o cuando el dispositivo en realidad no contiene engranajes, como en una transmisión continuamente variable . [2]
Además, un engranaje puede engranar con una parte dentada lineal, llamada cremallera , produciendo traslación en lugar de rotación.
Historia
Los primeros ejemplos de engranajes datan del siglo IV a. C. en China [3] (época de Zhan Guo - dinastía Zhou del Este tardío ), que se han conservado en el Museo Luoyang de la provincia de Henan, China . Los primeros engranajes conservados en Europa se encontraron en el mecanismo de Antikythera , un ejemplo de un dispositivo de engranajes muy temprano e intrincado, diseñado para calcular posiciones astronómicas . Su tiempo de construcción se estima en la actualidad entre el 150 y el 100 a. C. [4] Los engranajes aparecen en las obras conectados a Herón de Alejandría , en Egipto romano alrededor del año 50 dC, [5] pero se pueden remontar a la mecánica de la escuela de Alejandría en el siglo 3 aC Egipto ptolemaico , y se desarrollaron en gran medida por el griego el erudito Arquímedes (287-212 a. C.). [6]
El engranaje segmentario, que recibe / comunica movimiento alternativo desde / hacia una rueda dentada, que consiste en un sector de un engranaje / anillo circular que tiene engranajes en la periferia, [7] fue inventado por el ingeniero árabe Al-Jazari en 1206. [8] El El engranaje de gusano se inventó en el subcontinente indio , para su uso en desmotadoras de algodón , en algún momento durante los siglos XIII-XIV. [9] Es posible que se hayan utilizado engranajes diferenciales en algunos de los carros chinos que apuntan al sur , [10] pero el primer uso verificable de engranajes diferenciales fue el fabricante de relojes británico Joseph Williamson en 1720.
Ejemplos de aplicaciones de engranajes tempranos incluyen:
- 1386 EC: El reloj de la catedral de Salisbury : es el reloj mecánico con engranajes más antiguo del mundo que aún funciona.
- C. Siglos XIII-XIV: El engranaje helicoidal se inventó como parte de una desmotadora de algodón con rodillo en el subcontinente indio . [9]
- C. 1221 EC El astrolabio con engranajes se construyó en Isfahan y muestra la posición de la luna en el zodíaco y su fase , y el número de días desde la luna nueva. [11]
- C. 1206 CE: Al-Jazari inventó el engranaje segmentario como parte de un dispositivo de elevación de agua. [8]
- 725 d.C .: Los primeros relojes mecánicos con engranajes se construyeron en China .
- C. 200-265 d. C.: Ma Jun usó engranajes como parte de un carro que apuntaba al sur .
- Siglo II a.C .: El mecanismo de Antikythera
- En la naturaleza: en las patas traseras de las ninfas del insecto saltahojas Issus coleoptratus .
Etimología
La palabra engranaje es probablemente del nórdico antiguo gørvi (plural gørvar ) 'ropa, engranaje', relacionado con gøra , gørva 'hacer, construir, construir; poner en orden, preparar ', un verbo común en nórdico antiguo, "utilizado en una amplia gama de situaciones, desde escribir un libro hasta aderezar carne". En este contexto, el significado de 'rueda dentada en maquinaria' atestiguó por primera vez la década de 1520; el sentido mecánico específico de "partes mediante las cuales un motor comunica movimiento" es de 1814; específicamente de un vehículo (bicicleta, automóvil, etc.) en 1888. [12]
Un engranaje es un diente en una rueda. Del inglés medio cogge, del nórdico antiguo (compárese con el noruego kugg ('cog'), el sueco kugg , kugge ('cog, tooth')), del protogermánico * kuggō (compárese con el holandés kogge (' cogboat '), el alemán Kock ) , del protoindoeuropeo * gugā ('joroba, bola') (compárese con gugà lituano ('pomo, joroba, colina'), de PIE * gēw- ('doblar, arco'). [13] Se utilizó por primera vez c . 1300 en el sentido de 'una rueda que tiene dientes o piñones; finales del 14c.,' Diente en una rueda '; rueda dentada, principios del 15c. [14]
Históricamente, las ruedas dentadas eran dientes de madera en lugar de metal, y una rueda dentada consistía técnicamente en una serie de dientes de madera ubicados alrededor de una rueda de mortaja, cada diente formando un tipo de junta de espiga y mortaja "pasante" especializada . La rueda puede estar hecha de madera, hierro fundido u otro material. Los engranajes de madera se usaban anteriormente cuando no se podían cortar engranajes metálicos grandes, cuando el diente fundido no tenía ni siquiera aproximadamente la forma adecuada, o el tamaño de la rueda hacía que la fabricación no fuera práctica. [15]
Los engranajes a menudo estaban hechos de madera de arce . En 1967, la Thompson Manufacturing Company de Lancaster, New Hampshire todavía tenía un negocio muy activo en el suministro de decenas de miles de dientes de engranajes de arce por año, principalmente para su uso en fábricas de papel y molinos , algunos con más de 100 años. [16] Dado que un diente de madera realiza exactamente la misma función que un diente de metal fundido o mecanizado, la palabra se aplicó por extensión a ambos, y la distinción se ha perdido en general.
Comparación con los mecanismos de accionamiento
La relación definida que los dientes dan a los engranajes proporciona una ventaja sobre otras transmisiones (como las transmisiones de tracción y las correas trapezoidales ) en máquinas de precisión, como los relojes, que dependen de una relación de velocidad exacta. En los casos en que el impulsor y el seguidor están próximos, los engranajes también tienen una ventaja sobre otros accionamientos en el número reducido de piezas necesarias. La desventaja es que los engranajes son más costosos de fabricar y sus requisitos de lubricación pueden imponer un mayor costo operativo por hora.
Tipos
Engranajes externos versus internos
Un engranaje externo es uno con los dientes formados en la superficie externa de un cilindro o cono. Por el contrario, un engranaje interno es uno con los dientes formados en la superficie interna de un cilindro o cono. Para engranajes cónicos , un engranaje interno es uno con el lanzamiento ángulo superior a 90 grados. Los engranajes internos no provocan la inversión de la dirección del eje de salida. [17]
Estimular
Los engranajes rectos o de corte recto son el tipo de engranaje más simple. Consisten en un cilindro o disco con dientes que se proyectan radialmente. Aunque los dientes no son de lados rectos (pero generalmente de forma especial para lograr una relación de impulsión constante, principalmente involuta pero menos comúnmente cicloidal ), el borde de cada diente es recto y alineado paralelo al eje de rotación. Estos engranajes se engranan correctamente solo si están instalados en ejes paralelos. [18] Las cargas de los dientes no crean ningún empuje axial. Los engranajes rectos son excelentes a velocidades moderadas, pero tienden a ser ruidosos a altas velocidades. [19]
Helicoidal
Los engranajes helicoidales o "fijos en seco" ofrecen un refinamiento sobre los engranajes rectos. Los bordes de ataque de los dientes no son paralelos al eje de rotación, sino que están colocados en ángulo. Dado que el engranaje es curvo, este ángulo hace que el diente forme un segmento de hélice . Los engranajes helicoidales se pueden engranar en orientación paralela o cruzada . El primero se refiere a cuando los ejes son paralelos entre sí; esta es la orientación más común. En este último, los ejes no son paralelos y, en esta configuración, los engranajes a veces se conocen como "engranajes oblicuos".
Los dientes en ángulo se acoplan más gradualmente que los dientes de engranajes rectos, lo que hace que funcionen de manera más suave y silenciosa. [20] Con engranajes helicoidales paralelos, cada par de dientes primero hace contacto en un solo punto en un lado de la rueda dentada; una curva de contacto en movimiento luego crece gradualmente a lo largo de la cara del diente hasta un máximo, luego retrocede hasta que los dientes rompen el contacto en un solo punto en el lado opuesto. En los engranajes rectos, los dientes se encuentran repentinamente en una línea de contacto a lo largo de todo su ancho, lo que provoca estrés y ruido. Los engranajes rectos emiten un chirrido característico a altas velocidades. Por esta razón, los engranajes rectos se utilizan en aplicaciones de baja velocidad y en situaciones en las que el control del ruido no es un problema, y los engranajes helicoidales se utilizan en aplicaciones de alta velocidad, transmisión de gran potencia o donde la reducción del ruido es importante. [21] La velocidad se considera alta cuando la velocidad de la línea de paso supera los 25 m / s. [22]
Una desventaja de los engranajes helicoidales es el empuje resultante a lo largo del eje del engranaje, que debe ser acomodado por cojinetes de empuje apropiados . Sin embargo, este problema puede convertirse en una ventaja cuando se utiliza un engranaje en espiga o un engranaje helicoidal doble , que no tiene empuje axial y también proporciona autoalineación de los engranajes. Esto da como resultado un menor empuje axial que un engranaje recto comparable.
Una segunda desventaja de los engranajes helicoidales es también un mayor grado de fricción por deslizamiento entre los dientes de engranaje, que a menudo se soluciona con aditivos en el lubricante.
Engranajes sesgados
Para una configuración "cruzada" o "oblicua", los engranajes deben tener el mismo ángulo de presión y paso normal; sin embargo, el ángulo de la hélice y la orientación manual pueden ser diferentes. La relación entre los dos ejes se define en realidad por el ángulo o ángulos de hélice de los dos ejes y la orientación, como se define: [23]
- para engranajes de la misma mano,
- para engranajes de mano opuesta,
dónde es el ángulo de la hélice del engranaje. La configuración cruzada es menos sólida mecánicamente porque solo hay un punto de contacto entre los engranajes, mientras que en la configuración paralela hay un contacto de línea. [23]
Muy comúnmente, los engranajes helicoidales se utilizan con el ángulo de hélice de uno que tiene el negativo del ángulo de hélice del otro; tal par también podría denominarse que tiene una hélice a la derecha y una hélice a la izquierda de ángulos iguales. Los dos ángulos iguales pero opuestos se suman a cero: el ángulo entre ejes es cero, es decir, los ejes son paralelos . Cuando la suma o la diferencia (como se describe en las ecuaciones anteriores) no es cero, los ejes se cruzan . Para ejes cruzados en ángulos rectos, los ángulos de la hélice son de la misma mano porque deben sumar 90 grados. (Este es el caso de los engranajes de la ilustración anterior: engranan correctamente en la configuración cruzada: para la configuración paralela, uno de los ángulos de la hélice debe invertirse. Los engranajes ilustrados no pueden engranar con los ejes paralelos).
- Animación 3D de engranajes helicoidales (eje paralelo)
- Animación 3D de engranajes helicoidales (eje cruzado)
Helicoidal doble
Los engranajes helicoidales dobles superan el problema del empuje axial que presentan los engranajes helicoidales simples mediante el uso de un juego de dientes doble, inclinados en direcciones opuestas. Se puede pensar en un engranaje helicoidal doble como dos engranajes helicoidales espejados montados muy juntos en un eje común. Esta disposición anula el empuje axial neto, ya que cada mitad del engranaje empuja en la dirección opuesta, lo que resulta en una fuerza axial neta de cero. Esta disposición también puede eliminar la necesidad de cojinetes de empuje. Sin embargo, los engranajes helicoidales dobles son más difíciles de fabricar debido a su forma más complicada.
Los engranajes en espiga son un tipo especial de engranajes helicoidales. No tienen una ranura en el medio como lo tienen otros engranajes helicoidales dobles; los dos engranajes helicoidales reflejados se unen para que sus dientes formen una V. Esto también se puede aplicar a los engranajes cónicos , como en la transmisión final del Tipo A Citroën .
Para los dos posibles sentidos de giro, existen dos posibles disposiciones para los engranajes helicoidales o caras de engranajes orientados de forma opuesta. Un arreglo se llama estable y el otro inestable. En una disposición estable, las caras del engranaje helicoidal están orientadas de modo que cada fuerza axial se dirija hacia el centro del engranaje. En una disposición inestable, ambas fuerzas axiales se dirigen lejos del centro del engranaje. En cualquier disposición, la fuerza axial total (o neta ) en cada engranaje es cero cuando los engranajes están alineados correctamente. Si los engranajes se desalinean en la dirección axial, la disposición inestable genera una fuerza neta que puede conducir al desmontaje del tren de engranajes, mientras que la disposición estable genera una fuerza correctiva neta. Si se invierte la dirección de rotación, también se invierte la dirección de los empujes axiales, por lo que una configuración estable se vuelve inestable, y viceversa.
Stable double helical gears can be directly interchanged with spur gears without any need for different bearings.
Bevel
A bevel gear is shaped like a right circular cone with most of its tip cut off. When two bevel gears mesh, their imaginary vertices must occupy the same point. Their shaft axes also intersect at this point, forming an arbitrary non-straight angle between the shafts. The angle between the shafts can be anything except zero or 180 degrees. Bevel gears with equal numbers of teeth and shaft axes at 90 degrees are called miter (US) or mitre (UK) gears.
Spiral bevels
Spiral bevel gears can be manufactured as Gleason types (circular arc with non-constant tooth depth), Oerlikon and Curvex types (circular arc with constant tooth depth), Klingelnberg Cyclo-Palloid (Epicycloid with constant tooth depth) or Klingelnberg Palloid. Spiral bevel gears have the same advantages and disadvantages relative to their straight-cut cousins as helical gears do to spur gears. Straight bevel gears are generally used only at speeds below 5 m/s (1000 ft/min), or, for small gears, 1000 r.p.m.[24]
Note: The cylindrical gear tooth profile corresponds to an involute, but the bevel gear tooth profile to an octoid. All traditional bevel gear generators (like Gleason, Klingelnberg, Heidenreich & Harbeck, WMW Modul) manufacture bevel gears with an octoidal tooth profile. IMPORTANT: For 5-axis milled bevel gear sets it is important to choose the same calculation / layout like the conventional manufacturing method. Simplified calculated bevel gears on the basis of an equivalent cylindrical gear in normal section with an involute tooth form show a deviant tooth form with reduced tooth strength by 10-28% without offset and 45% with offset [Diss. Hünecke, TU Dresden]. Furthermore, the "involute bevel gear sets" cause more noise.
Hypoid
Hypoid gears resemble spiral bevel gears except the shaft axes do not intersect. The pitch surfaces appear conical but, to compensate for the offset shaft, are in fact hyperboloids of revolution.[25][26] Hypoid gears are almost always designed to operate with shafts at 90 degrees. Depending on which side the shaft is offset to, relative to the angling of the teeth, contact between hypoid gear teeth may be even smoother and more gradual than with spiral bevel gear teeth, but also have a sliding action along the meshing teeth as it rotates and therefore usually require some of the most viscous types of gear oil to avoid it being extruded from the mating tooth faces, the oil is normally designated HP (for hypoid) followed by a number denoting the viscosity. Also, the pinion can be designed with fewer teeth than a spiral bevel pinion, with the result that gear ratios of 60:1 and higher are feasible using a single set of hypoid gears.[27] This style of gear is most common in motor vehicle drive trains, in concert with a differential. Whereas a regular (nonhypoid) ring-and-pinion gear set is suitable for many applications, it is not ideal for vehicle drive trains because it generates more noise and vibration than a hypoid does. Bringing hypoid gears to market for mass-production applications was an engineering improvement of the 1920s.
Crown
Crown gears or contrate gears are a particular form of bevel gear whose teeth project at right angles to the plane of the wheel; in their orientation the teeth resemble the points on a crown. A crown gear can only mesh accurately with another bevel gear, although crown gears are sometimes seen meshing with spur gears. A crown gear is also sometimes meshed with an escapement such as found in mechanical clocks.
Worm
Worms resemble screws. A worm is meshed with a worm wheel, which looks similar to a spur gear.
Worm-and-gear sets are a simple and compact way to achieve a high torque, low speed gear ratio. For example, helical gears are normally limited to gear ratios of less than 10:1 while worm-and-gear sets vary from 10:1 to 500:1.[28] A disadvantage is the potential for considerable sliding action, leading to low efficiency.[29]
A worm gear is a species of helical gear, but its helix angle is usually somewhat large (close to 90 degrees) and its body is usually fairly long in the axial direction. These attributes give it screw like qualities. The distinction between a worm and a helical gear is that at least one tooth persists for a full rotation around the helix. If this occurs, it is a 'worm'; if not, it is a 'helical gear'. A worm may have as few as one tooth. If that tooth persists for several turns around the helix, the worm appears, superficially, to have more than one tooth, but what one in fact sees is the same tooth reappearing at intervals along the length of the worm. The usual screw nomenclature applies: a one-toothed worm is called single thread or single start; a worm with more than one tooth is called multiple thread or multiple start. The helix angle of a worm is not usually specified. Instead, the lead angle, which is equal to 90 degrees minus the helix angle, is given.
In a worm-and-gear set, the worm can always drive the gear. However, if the gear attempts to drive the worm, it may or may not succeed. Particularly if the lead angle is small, the gear's teeth may simply lock against the worm's teeth, because the force component circumferential to the worm is not sufficient to overcome friction. In traditional music boxes, however, the gear drives the worm, which has a large helix angle. This mesh drives the speed-limiter vanes which are mounted on the worm shaft.
Worm-and-gear sets that do lock are called self locking, which can be used to advantage, as when it is desired to set the position of a mechanism by turning the worm and then have the mechanism hold that position. An example is the machine head found on some types of stringed instruments.
If the gear in a worm-and-gear set is an ordinary helical gear only a single point of contact is achieved.[27][30] If medium to high power transmission is desired, the tooth shape of the gear is modified to achieve more intimate contact by making both gears partially envelop each other. This is done by making both concave and joining them at a saddle point; this is called a cone-drive[31] or "Double enveloping".
Worm gears can be right or left-handed, following the long-established practice for screw threads.[17]
- 3D Animation of a worm-gear set
Non-circular
Non-circular gears are designed for special purposes. While a regular gear is optimized to transmit torque to another engaged member with minimum noise and wear and maximum efficiency, a non-circular gear's main objective might be ratio variations, axle displacement oscillations and more. Common applications include textile machines, potentiometers and continuously variable transmissions.
Rack and pinion
A rack is a toothed bar or rod that can be thought of as a sector gear with an infinitely large radius of curvature. Torque can be converted to linear force by meshing a rack with a round gear called a pinion: the pinion turns, while the rack moves in a straight line. Such a mechanism is used in automobiles to convert the rotation of the steering wheel into the left-to-right motion of the tie rod(s).
Racks also feature in the theory of gear geometry, where, for instance, the tooth shape of an interchangeable set of gears may be specified for the rack (infinite radius), and the tooth shapes for gears of particular actual radii are then derived from that. The rack and pinion gear type is also used in a rack railway.
Epicyclic
In epicyclic gearing, one or more of the gear axes moves. Examples are sun and planet gearing (see below), cycloidal drive, automatic transmissions, and mechanical differentials.
Sun and planet
Sun and planet gearing is a method of converting reciprocating motion into rotary motion that was used in steam engines. James Watt used it on his early steam engines to get around the patent on the crank, but it also provided the advantage of increasing the flywheel speed so Watt could use a lighter flywheel.
In the illustration, the sun is yellow, the planet red, the reciprocating arm is blue, the flywheel is green and the driveshaft is gray.
Harmonic gear
A harmonic gear or strain wave gear is a specialized gearing mechanism often used in industrial motion control, robotics and aerospace for its advantages over traditional gearing systems, including lack of backlash, compactness and high gear ratios.
Though the diagram does not demonstrate the correct configuration, it is a "timing gear," conventionally with far more teeth than a traditional gear to ensure a higher degree of precision.
Cage gear
A cage gear, also called a lantern gear or lantern pinion, has cylindrical rods for teeth, parallel to the axle and arranged in a circle around it, much as the bars on a round bird cage or lantern. The assembly is held together by disks at each end, into which the tooth rods and axle are set. Cage gears are more efficient than solid pinions,[citation needed] and dirt can fall through the rods rather than becoming trapped and increasing wear. They can be constructed with very simple tools as the teeth are not formed by cutting or milling, but rather by drilling holes and inserting rods.
Sometimes used in clocks, the cage gear should always be driven by a gearwheel, not used as the driver. The cage gear was not initially favoured by conservative clock makers. It became popular in turret clocks where dirty working conditions were most commonplace. Domestic American clock movements often used them.
Cycloidal gear
Magnetic gear
All cogs of each gear component of magnetic gears act as a constant magnet with periodic alternation of opposite magnetic poles on mating surfaces. Gear components are mounted with a backlash capability similar to other mechanical gearings. Although they cannot exert as much force as a traditional gear due to limits on magnetic field strength, such gears work without touching and so are immune to wear, have very low noise, no power losses from friction and can slip without damage making them very reliable.[32] They can be used in configurations that are not possible for gears that must be physically touching and can operate with a non-metallic barrier completely separating the driving force from the load. The magnetic coupling can transmit force into a hermetically sealed enclosure without using a radial shaft seal, which may leak.
Nomenclatura
General
- Rotational frequency, n
- Measured in rotation over time, such as revolutions per minute (RPM or rpm).
- Angular frequency, ω
- Measured in radians/second. 1 RPM = 2π rad/minute = π/30 rad/second.
- Number of teeth, N
- How many teeth a gear has, an integer. In the case of worms, it is the number of thread starts that the worm has.
- Gear, wheel
- The larger of two interacting gears or a gear on its own.
- Pinion
- The smaller of two interacting gears.
- Path of contact
- Path followed by the point of contact between two meshing gear teeth.
- Line of action, pressure line
- Line along which the force between two meshing gear teeth is directed. It has the same direction as the force vector. In general, the line of action changes from moment to moment during the period of engagement of a pair of teeth. For involute gears, however, the tooth-to-tooth force is always directed along the same line—that is, the line of action is constant. This implies that for involute gears the path of contact is also a straight line, coincident with the line of action—as is indeed the case.
- Axis
- Axis of revolution of the gear; center line of the shaft.
- Pitch point
- Point where the line of action crosses a line joining the two gear axes.
- Pitch circle, pitch line
- Circle centered on and perpendicular to the axis, and passing through the pitch point. A predefined diametral position on the gear where the circular tooth thickness, pressure angle and helix angles are defined.
- Pitch diameter, d
- A predefined diametral position on the gear where the circular tooth thickness, pressure angle and helix angles are defined. The standard pitch diameter is a design dimension and cannot be measured, but is a location where other measurements are made. Its value is based on the number of teeth ( N), the normal module ( mn; or normal diametral pitch, Pd), and the helix angle ( ):
- in metric units or in imperial units. [33]
- Module or modulus, m
- Since it is impractical to calculate circular pitch with irrational numbers, mechanical engineers usually use a scaling factor that replaces it with a regular value instead. This is known as the module or modulus of the wheel and is simply defined as:
- where m is the module and p the circular pitch. The units of module are customarily millimeters; an English Module is sometimes used with the units of inches. When the diametral pitch, DP, is in English units,
- in conventional metric units.
- The distance between the two axis becomes:
- where a is the axis distance, z 1 and z 2 are the number of cogs (teeth) for each of the two wheels (gears). These numbers (or at least one of them) is often chosen among primes to create an even contact between every cog of both wheels, and thereby avoid unnecessary wear and damage. An even uniform gear wear is achieved by ensuring the tooth counts of the two gears meshing together are relatively prime to each other; this occurs when the greatest common divisor (GCD) of each gear tooth count equals 1, e.g. GCD(16,25)=1; if a 1:1 gear ratio is desired a relatively prime gear may be inserted in between the two gears; this maintains the 1:1 ratio but reverses the gear direction; a second relatively prime gear could also be inserted to restore the original rotational direction while maintaining uniform wear with all 4 gears in this case. Mechanical engineers, at least in continental Europe, usually use the module instead of circular pitch. The module, just like the circular pitch, can be used for all types of cogs, not just evolvent based straight cogs. [34]
- Operating pitch diameters
- Diameters determined from the number of teeth and the center distance at which gears operate. [17] Example for pinion:
- Pitch surface
- In cylindrical gears, cylinder formed by projecting a pitch circle in the axial direction. More generally, the surface formed by the sum of all the pitch circles as one moves along the axis. For bevel gears it is a cone.
- Angle of action
- Angle with vertex at the gear center, one leg on the point where mating teeth first make contact, the other leg on the point where they disengage.
- Arc of action
- Segment of a pitch circle subtended by the angle of action.
- Pressure angle,
- The complement of the angle between the direction that the teeth exert force on each other, and the line joining the centers of the two gears. For involute gears, the teeth always exert force along the line of action, which, for involute gears, is a straight line; and thus, for involute gears, the pressure angle is constant.
- Outside diameter,
- Diameter of the gear, measured from the tops of the teeth.
- Root diameter
- Diameter of the gear, measured at the base of the tooth.
- Addendum, a
- Radial distance from the pitch surface to the outermost point of the tooth.
- Dedendum, b
- Radial distance from the depth of the tooth trough to the pitch surface.
- Whole depth,
- The distance from the top of the tooth to the root; it is equal to addendum plus dedendum or to working depth plus clearance.
- Clearance
- Distance between the root circle of a gear and the addendum circle of its mate.
- Working depth
- Depth of engagement of two gears, that is, the sum of their operating addendums.
- Circular pitch, p
- Distance from one face of a tooth to the corresponding face of an adjacent tooth on the same gear, measured along the pitch circle.
- Diametral pitch, DP
- Ratio of the number of teeth to the pitch diameter. Could be measured in teeth per inch or teeth per centimeter, but conventionally has units of per inch of diameter. Where the module, m, is in metric units
- in English units
- Base circle
- In involute gears, the tooth profile is generated by the involute of the base circle. The radius of the base circle is somewhat smaller than that of the pitch circle
- Base pitch, normal pitch,
- In involute gears, distance from one face of a tooth to the corresponding face of an adjacent tooth on the same gear, measured along the base circle
- Interference
- Contact between teeth other than at the intended parts of their surfaces
- Interchangeable set
- A set of gears, any of which mates properly with any other
Helical gear
- Helix angle,
- the Angle between a tangent to the helix and the gear axis. It is zero in the limiting case of a spur gear, albeit it can considered as the hypotenuse angle as well.
- Normal circular pitch,
- Circular pitch in the plane normal to the teeth.
- Transverse circular pitch, p
- Circular pitch in the plane of rotation of the gear. Sometimes just called "circular pitch".
Several other helix parameters can be viewed either in the normal or transverse planes. The subscript n usually indicates the normal.
Worm gear
- Lead
- Distance from any point on a thread to the corresponding point on the next turn of the same thread, measured parallel to the axis.
- Linear pitch, p
- Distance from any point on a thread to the corresponding point on the adjacent thread, measured parallel to the axis. For a single-thread worm, lead and linear pitch are the same.
- Lead angle,
- Angle between a tangent to the helix and a plane perpendicular to the axis. Note that the complement of the helix angle is usually given for helical gears.
- Pitch diameter,
- Same as described earlier in this list. Note that for a worm it is still measured in a plane perpendicular to the gear axis, not a tilted plane.
Subscript w denotes the worm, subscript g denotes the gear.
Tooth contact
Line of contact
Path of action
Line of action
Plane of action
Lines of contact (helical gear)
Arc of action
Length of action
Limit diameter
Face advance
Zone of action
- Point of contact
- Any point at which two tooth profiles touch each other.
- Line of contact
- A line or curve along which two tooth surfaces are tangent to each other.
- Path of action
- The locus of successive contact points between a pair of gear teeth, during the phase of engagement. For conjugate gear teeth, the path of action passes through the pitch point. It is the trace of the surface of action in the plane of rotation.
- Line of action
- The path of action for involute gears. It is the straight line passing through the pitch point and tangent to both base circles.
- Surface of action
- The imaginary surface in which contact occurs between two engaging tooth surfaces. It is the summation of the paths of action in all sections of the engaging teeth.
- Plane of action
- The surface of action for involute, parallel axis gears with either spur or helical teeth. It is tangent to the base cylinders.
- Zone of action (contact zone)
- For involute, parallel-axis gears with either spur or helical teeth, is the rectangular area in the plane of action bounded by the length of action and the effective face width.
- Path of contact
- The curve on either tooth surface along which theoretical single point contact occurs during the engagement of gears with crowned tooth surfaces or gears that normally engage with only single point contact.
- Length of action
- The distance on the line of action through which the point of contact moves during the action of the tooth profile.
- Arc of action, Q t
- The arc of the pitch circle through which a tooth profile moves from the beginning to the end of contact with a mating profile.
- Arc of approach, Q a
- The arc of the pitch circle through which a tooth profile moves from its beginning of contact until the point of contact arrives at the pitch point.
- Arc of recess, Q r
- The arc of the pitch circle through which a tooth profile moves from contact at the pitch point until contact ends.
- Contact ratio, m c, ε
- The number of angular pitches through which a tooth surface rotates from the beginning to the end of contact. In a simple way, it can be defined as a measure of the average number of teeth in contact during the period during which a tooth comes and goes out of contact with the mating gear.
- Transverse contact ratio, m p, ε α
- The contact ratio in a transverse plane. It is the ratio of the angle of action to the angular pitch. For involute gears it is most directly obtained as the ratio of the length of action to the base pitch.
- Face contact ratio, m F, ε β
- The contact ratio in an axial plane, or the ratio of the face width to the axial pitch. For bevel and hypoid gears it is the ratio of face advance to circular pitch.
- Total contact ratio, m t, ε γ
- The sum of the transverse contact ratio and the face contact ratio.
- Modified contact ratio, m o
- For bevel gears, the square root of the sum of the squares of the transverse and face contact ratios.
- Limit diameter
- Diameter on a gear at which the line of action intersects the maximum (or minimum for internal pinion) addendum circle of the mating gear. This is also referred to as the start of active profile, the start of contact, the end of contact, or the end of active profile.
- Start of active profile (SAP)
- Intersection of the limit diameter and the involute profile.
- Face advance
- Distance on a pitch circle through which a helical or spiral tooth moves from the position at which contact begins at one end of the tooth trace on the pitch surface to the position where contact ceases at the other end.
Tooth thickness
Tooth thickness
Thickness relationships
Chordal thickness
Tooth thickness measurement over pins
Span measurement
Long and short addendum teeth
- Circular thickness
- Length of arc between the two sides of a gear tooth, on the specified datum circle.
- Transverse circular thickness
- Circular thickness in the transverse plane.
- Normal circular thickness
- Circular thickness in the normal plane. In a helical gear it may be considered as the length of arc along a normal helix.
- Axial thickness
- In helical gears and worms, tooth thickness in an axial cross section at the standard pitch diameter.
- Base circular thickness
- In involute teeth, length of arc on the base circle between the two involute curves forming the profile of a tooth.
- Normal chordal thickness
- Length of the chord that subtends a circular thickness arc in the plane normal to the pitch helix. Any convenient measuring diameter may be selected, not necessarily the standard pitch diameter.
- Chordal addendum (chordal height)
- Height from the top of the tooth to the chord subtending the circular thickness arc. Any convenient measuring diameter may be selected, not necessarily the standard pitch diameter.
- Profile shift
- Displacement of the basic rack datum line from the reference cylinder, made non-dimensional by dividing by the normal module. It is used to specify the tooth thickness, often for zero backlash.
- Rack shift
- Displacement of the tool datum line from the reference cylinder, made non-dimensional by dividing by the normal module. It is used to specify the tooth thickness.
- Measurement over pins
- Measurement of the distance taken over a pin positioned in a tooth space and a reference surface. The reference surface may be the reference axis of the gear, a datum surface or either one or two pins positioned in the tooth space or spaces opposite the first. This measurement is used to determine tooth thickness.
- Span measurement
- Measurement of the distance across several teeth in a normal plane. As long as the measuring device has parallel measuring surfaces that contact on an unmodified portion of the involute, the measurement wis along a line tangent to the base cylinder. It is used to determine tooth thickness.
- Modified addendum teeth
- Teeth of engaging gears, one or both of which have non-standard addendum.
- Full-depth teeth
- Teeth in which the working depth equals 2.000 divided by the normal diametral pitch.
- Stub teeth
- Teeth in which the working depth is less than 2.000 divided by the normal diametral pitch.
- Equal addendum teeth
- Teeth in which two engaging gears have equal addendums.
- Long and short-addendum teeth
- Teeth in which the addendums of two engaging gears are unequal.
Pitch
Pitch is the distance between a point on one tooth and the corresponding point on an adjacent tooth.[17] It is a dimension measured along a line or curve in the transverse, normal, or axial directions. The use of the single word pitch without qualification may be ambiguous, and for this reason it is preferable to use specific designations such as transverse circular pitch, normal base pitch, axial pitch.
Pitch
Tooth pitch
Base pitch relationships
Principal pitches
- Circular pitch, p
- Arc distance along a specified pitch circle or pitch line between corresponding profiles of adjacent teeth.
- Transverse circular pitch, pt
- Circular pitch in the transverse plane.
- Normal circular pitch, pn, pe
- Circular pitch in the normal plane, and also the length of the arc along the normal pitch helix between helical teeth or threads.
- Axial pitch, px
- Linear pitch in an axial plane and in a pitch surface. In helical gears and worms, axial pitch has the same value at all diameters. In gearing of other types, axial pitch may be confined to the pitch surface and may be a circular measurement. The term axial pitch is preferred to the term linear pitch. The axial pitch of a helical worm and the circular pitch of its worm gear are the same.
- Normal base pitch, pN, pbn
- An involute helical gear is the base pitch in the normal plane. It is the normal distance between parallel helical involute surfaces on the plane of action in the normal plane, or is the length of arc on the normal base helix. It is a constant distance in any helical involute gear.
- Transverse base pitch, pb, pbt
- In an involute gear, the pitch is on the base circle or along the line of action. Corresponding sides of involute gear teeth are parallel curves, and the base pitch is the constant and fundamental distance between them along a common normal in a transverse plane.
- Diametral pitch (transverse), Pd
- Ratio of the number of teeth to the standard pitch diameter in inches.
- Normal diametrical pitch, Pnd
- Value of diametrical pitch in a normal plane of a helical gear or worm.
- Angular pitch, θ N, τ
- Angle subtended by the circular pitch, usually expressed in radians.
- degrees or radians
Reacción
Backlash is the error in motion that occurs when gears change direction. It exists because there is always some gap between the trailing face of the driving tooth and the leading face of the tooth behind it on the driven gear, and that gap must be closed before force can be transferred in the new direction. The term "backlash" can also be used to refer to the size of the gap, not just the phenomenon it causes; thus, one could speak of a pair of gears as having, for example, "0.1 mm of backlash." A pair of gears could be designed to have zero backlash, but this would presuppose perfection in manufacturing, uniform thermal expansion characteristics throughout the system, and no lubricant. Therefore, gear pairs are designed to have some backlash. It is usually provided by reducing the tooth thickness of each gear by half the desired gap distance. In the case of a large gear and a small pinion, however, the backlash is usually taken entirely off the gear and the pinion is given full sized teeth. Backlash can also be provided by moving the gears further apart. The backlash of a gear train equals the sum of the backlash of each pair of gears, so in long trains backlash can become a problem.
For situations that require precision, such as instrumentation and control, backlash can be minimized through one of several techniques. For instance, the gear can be split along a plane perpendicular to the axis, one half fixed to the shaft in the usual manner, the other half placed alongside it, free to rotate about the shaft, but with springs between the two-halves providing relative torque between them, so that one achieves, in effect, a single gear with expanding teeth. Another method involves tapering the teeth in the axial direction and letting the gear slide in the axial direction to take up slack.
Cambio de marchas
In some machines (e.g., automobiles) it is necessary to alter the gear ratio to suit the task, a process known as gear shifting or changing gear. There are several ways of shifting gears, for example:
- Manual transmission
- Automatic transmission
- Derailleur gears, which are actually sprockets in combination with a roller chain
- Hub gears (also called epicyclic gearing or sun-and-planet gears)
There are several outcomes of gear shifting in motor vehicles. In the case of vehicle noise emissions, there are higher sound levels emitted when the vehicle is engaged in lower gears. The design life of the lower ratio gears is shorter, so cheaper gears may be used, which tend to generate more noise due to smaller overlap ratio and a lower mesh stiffness etc. than the helical gears used for the high ratios. This fact has been used to analyze vehicle-generated sound since the late 1960s, and has been incorporated into the simulation of urban roadway noise and corresponding design of urban noise barriers along roadways.[35]
Perfil de diente
Profile of a spur gear
Undercut
A profile is one side of a tooth in a cross section between the outside circle and the root circle. Usually a profile is the curve of intersection of a tooth surface and a plane or surface normal to the pitch surface, such as the transverse, normal, or axial plane.
The fillet curve (root fillet) is the concave portion of the tooth profile where it joins the bottom of the tooth space.2
As mentioned near the beginning of the article, the attainment of a nonfluctuating velocity ratio is dependent on the profile of the teeth. Friction and wear between two gears is also dependent on the tooth profile. There are a great many tooth profiles that provide constant velocity ratios. In many cases, given an arbitrary tooth shape, it is possible to develop a tooth profile for the mating gear that provides a constant velocity ratio. However, two constant velocity tooth profiles are the most commonly used in modern times: the cycloid and the involute. The cycloid was more common until the late 1800s. Since then, the involute has largely superseded it, particularly in drive train applications. The cycloid is in some ways the more interesting and flexible shape; however the involute has two advantages: it is easier to manufacture, and it permits the center-to-center spacing of the gears to vary over some range without ruining the constancy of the velocity ratio. Cycloidal gears only work properly if the center spacing is exactly right. Cycloidal gears are still used in mechanical clocks.
An undercut is a condition in generated gear teeth when any part of the fillet curve lies inside of a line drawn tangent to the working profile at its point of juncture with the fillet. Undercut may be deliberately introduced to facilitate finishing operations. With undercut the fillet curve intersects the working profile. Without undercut the fillet curve and the working profile have a common tangent.
Materiales de engranajes
Numerous nonferrous alloys, cast irons, powder-metallurgy and plastics are used in the manufacture of gears. However, steels are most commonly used because of their high strength-to-weight ratio and low cost. Plastic is commonly used where cost or weight is a concern. A properly designed plastic gear can replace steel in many cases because it has many desirable properties, including dirt tolerance, low speed meshing, the ability to "skip" quite well[36] and the ability to be made with materials that don't need additional lubrication. Manufacturers have used plastic gears to reduce costs in consumer items including copy machines, optical storage devices, cheap dynamos, consumer audio equipment, servo motors, and printers. Another advantage of the use of plastics, formerly (such as in the 1980s), was the reduction of repair costs for certain expensive machines. In cases of severe jamming (as of the paper in a printer), the plastic gear teeth would be torn free of their substrate, allowing the drive mechanism to then spin freely (instead of damaging itself by straining against the jam). This use of "sacrificial" gear teeth avoided destroying the much more expensive motor and related parts. This method has been superseded, in more recent designs, by the use of clutches and torque- or current-limited motors.
Parcelas estándar y el sistema de módulos
Although gears can be made with any pitch, for convenience and interchangeability standard pitches are frequently used. Pitch is a property associated with linear dimensions and so differs whether the standard values are in the imperial (inch) or metric systems. Using inch measurements, standard diametral pitch values with units of "per inch" are chosen; the diametrical pitch is the number of teeth on a gear of one inch pitch diameter. Common standard values for spur gears are 3, 4, 5, 6, 8, 10, 12, 16, 20, 24, 32, 48, 64, 72, 80, 96, 100, 120, and 200.[37] Certain standard pitches such as 1/10 and 1/20 in inch measurements, which mesh with linear rack, are actually (linear) circular pitch values with units of "inches"[37]
When gear dimensions are in the metric system the pitch specification is generally in terms of module or modulus, which is effectively a length measurement across the pitch diameter. The term module is understood to mean the pitch diameter in millimetres divided by the number of teeth. When the module is based upon inch measurements, it is known as the English module to avoid confusion with the metric module. Module is a direct dimension, unlike diametrical pitch, which is an inverse dimension ("threads per inch"). Thus, if the pitch diameter of a gear is 40 mm and the number of teeth 20, the module is 2, which means that there are 2 mm of pitch diameter for each tooth.[38] The preferred standard module values are 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.8, 1.0, 1.25, 1.5, 2.0, 2.5, 3, 4, 5, 6, 8, 10, 12, 16, 20, 25, 32, 40 and 50.[39]
Fabricar
As of 2014, an estimated 80% of all gearing produced worldwide is produced by net shape molding. Molded gearing is usually either powder metallurgy or plastic.[40] Many gears are done when they leave the mold (including injection molded plastic and die cast metal gears), but powdered metal gears require sintering and sand castings or investment castings require gear cutting or other machining to finish them. The most common form of gear cutting is hobbing, but gear shaping, milling, and broaching also exist. 3D printing as a production method is expanding rapidly. For metal gears in the transmissions of cars and trucks, the teeth are heat treated to make them hard and more wear resistant while leaving the core soft and tough. For large gears that are prone to warp, a quench press is used.
Modelo de engranajes en la física moderna.
Modern physics adopted the gear model in different ways. In the nineteenth century, James Clerk Maxwell developed a model of electromagnetism in which magnetic field lines were rotating tubes of incompressible fluid. Maxwell used a gear wheel and called it an "idle wheel" to explain the electric current as a rotation of particles in opposite directions to that of the rotating field lines.[41]
More recently, quantum physics uses "quantum gears" in their model. A group of gears can serve as a model for several different systems, such as an artificially constructed nanomechanical device or a group of ring molecules.[42]
The three wave hypothesis compares the wave–particle duality to a bevel gear.[43]
Mecanismo de engranajes en el mundo natural.
The gear mechanism was previously considered exclusively artificial, but as early as 1957, gears had been recognized in the hind legs of various species of planthoppers[44] and scientists from the University of Cambridge characterized their functional significance in 2013 by doing high-speed photography of the nymphs of Issus coleoptratus at Cambridge University.[45][46] These gears are found only in the nymph forms of all planthoppers, and are lost during the final molt to the adult stage.[47] In I. coleoptratus, each leg has a 400-micrometer strip of teeth, pitch radius 200 micrometers, with 10 to 12 fully interlocking spur-type gear teeth, including filleted curves at the base of each tooth to reduce the risk of shearing.[48] The joint rotates like mechanical gears, and synchronizes Issus's hind legs when it jumps to within 30 microseconds, preventing yaw rotation.[49][50][45] The gears aren't connected all the time. One is located on each of the juvenile insect's hind legs, and when it prepares to jump, the two sets of teeth lock together. As a result, the legs move in almost perfect unison, giving the insect more power as the gears rotate to their stopping point and then unlock.[49]
Ver también
- Gear box
- Sprocket
- Differential
- Superposition principle
- Kinematic chain
Referencias
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- ^ "Transmission Basics". HowStuffWorks.
- ^ Derek J. de Solla Price, On the Origin of Clockwork, Perpetual Motion Devices, and the Compass, p.84
- ^ "The Antikythera Mechanism Research Project: Why is it so important?". Archived from the original on 4 May 2012. Retrieved 10 January 2011.
The Mechanism is thought to date from between 150 and 100 BC
- ^ Norton 2004, p. 462
- ^ Lewis, M. J. T. (1993). "Gearing in the Ancient World". Endeavour. 17 (3): 110–115. doi:10.1016/0160-9327(93)90099-O.
- ^ "Segment gear". thefreedictionary.com. Retrieved 20 September 2018.
- ^ a b Donald Hill (2012), The Book of Knowledge of Ingenious Mechanical Devices, page 273, Springer Science + Business Media
- ^ a b Irfan Habib, Economic History of Medieval India, 1200-1500, page 53, Pearson Education
- ^ Joseph Needham (1986). Science and Civilization in China: Volume 4, Part 2, page 298. Taipei: Caves Books, Ltd.
- ^ "Astrolabe By Muhammad Ibn Abi Bakr Al Isfahani".
- ^ "gear (n.)". Etymonline. Retrieved 13 February 2020.
- ^ "Etymology 1: Cog (noun)". Wiktionary. Retrieved 29 July 2019.
- ^ "cog (n.)". Etymonline. Retrieved 13 February 2020.
- ^ Grant, George B. (1893). A Treatise on Gear Wheels (6th, illus. ed.). Lexington, MA; Philadelphia, PA: George B. Grant. p. 21.
- ^ Radzevich, Stephen P. (2012). Dudley's Handbook of Practical Gear Design and Manufacture (PDF) (2nd ed.). Boca Raton, FL.: CRC Press, an imprint of Taylor & Francis Group. pp. 691, 702.
- ^ a b c d American Gear Manufacturers Association; American National Standards Institute, Gear Nomenclature, Definitions of Terms with Symbols (ANSI/AGMA 1012-G05 ed.), American Gear Manufacturers Association
- ^ "How Gears Work". howstuffworks.com. 16 November 2000. Retrieved 20 September 2018.
- ^ Machinery's Handbook. New York: Industrial Press. 2012. pp. 2125. ISBN 978-0-8311-2900-2.
- ^ Khurmi, R. S., Theory of Machines, S.CHAND
- ^ Schunck, Richard, "Minimizing gearbox noise inside and outside the box", Motion System Design.
- ^ Vallance & Doughtie 1964, p. 281
- ^ a b Helical gears, archived from the original on 26 June 2009, retrieved 15 June 2009.
- ^ McGraw-Hill 2007, p. 742.
- ^ Canfield, Stephen (1997), "Gear Types", Dynamics of Machinery, Tennessee Tech University, Department of Mechanical Engineering, ME 362 lecture notes, archived from the original on 29 August 2008.
- ^ Hilbert, David; Cohn-Vossen, Stephan (1952), Geometry and the Imagination (2nd ed.), New York: Chelsea, p. 287, ISBN 978-0-8284-1087-8.
- ^ a b McGraw-Hill 2007, p. 743.
- ^ Vallance & Doughtie 1964, p. 287.
- ^ Vallance & Doughtie 1964, pp. 280, 296.
- ^ Vallance & Doughtie 1964, p. 290.
- ^ McGraw-Hill 2007, p. 744
- ^ Kravchenko A.I., Bovda A.M. Gear with magnetic couple. Pat. of Ukraine N. 56700 – Bul. N. 2, 2011 – F16H 49/00.
- ^ ISO/DIS 21771:2007 : "Gears – Cylindrical Involute Gears and Gear Pairs – Concepts and Geometry", International Organization for Standardization, (2007)
- ^ Gunnar Dahlvig (1982), "Construction elements and machine construction", Konstruktionselement och maskinbyggnad (in Swedish), 7, ISBN 978-9140115546
- ^ Hogan, C. Michael; Latshaw, Gary L. (21–23 May 1973). The Relationship Between Highway Planning and Urban Noise. Proceedings of the ASCE, Urban Transportation Division Specialty Conference. Chicago, Illinois: American Society of Civil Engineers, Urban Transportation Division.
- ^ Smith, Zan (2000), "Plastic gears are more reliable when engineers account for material properties and manufacturing processes during design.", Motion System Design.
- ^ a b "W. M. Berg Gear Reference Guide" (PDF). Archived from the original (PDF) on 21 April 2015.
- ^ Oberg, E.; Jones, F. D.; Horton, H. L.; Ryffell, H. H. (2000), Machinery's Handbook (26th ed.), Industrial Press, p. 2649, ISBN 978-0-8311-2666-7.
- ^ "Elements of metric gear technology" (PDF).
- ^ Fred Eberle (August 2014). "Materials Matter". Gear Solutions: 22.
- ^ Siegel, Daniel M. (1991). Innovation in Maxwell's Electromagnetic Theory: Molecular Vortices, Displacement Current, and Light. University of Chicago Press. ISBN 978-0521353656.
- ^ MacKinnon, Angus (2002). "Quantum Gears: A Simple Mechanical System in the Quantum Regime". Nanotechnology. 13 (5): 678–681. arXiv:cond-mat/0205647. Bibcode:2002Nanot..13..678M. doi:10.1088/0957-4484/13/5/328. S2CID 14994774.
- ^ Sanduk, M. I. (2007). "Does the Three Wave Hypothesis Imply Hidden Structure?" (PDF). Apeiron. 14 (2): 113–125. Bibcode:2007Apei...14..113S.
- ^ Sander, K. (1957), "Bau und Funktion des Sprungapparates von Pyrilla perpusilla WALKER (Homoptera - Fulgoridae)", Zool. Jb. Jena (Anat.) (in German), 75: 383–388
- ^ a b Burrows, Malcolm; Sutton, Gregory (13 September 2013). "Interacting Gears Synchronize Propulsive Leg Movements in a Jumping Insect". Science. 341 (6151): 1254–1256. doi:10.1126/science.1240284. hdl:1983/69cf1502-217a-4dca-a0d3-f8b247794e92. PMID 24031019. S2CID 24640726.
- ^ Herkewitz, William (12 September 2013), "The First Gear Discovered in Nature", Popular Mechanics
- ^ Lee, Jane J. (12 September 2013), "Insects Use Gears in Hind Legs to Jump", National Geographic
- ^ Stromberg, Joseph (12 September 2013), "This Insect Has The Only Mechanical Gears Ever Found in Nature", Smithsonian Magazine, retrieved 18 November 2020
- ^ a b Robertson, Adi (12 September 2013). "The first-ever naturally occurring gears are found on an insect's legs". The Verge. Retrieved 14 September 2013.
- ^ Functioning 'mechanical gears' seen in nature for the first time, PHYS.ORG, Cambridge University
Bibliography
- McGraw-Hill (2007), McGraw-Hill Encyclopedia of Science and Technology (10th ed.), McGraw-Hill Professional, ISBN 978-0-07-144143-8.
- Norton, Robert L. (2004), Design of Machinery (3rd ed.), McGraw-Hill Professional, ISBN 978-0-07-121496-4.
- Vallance, Alex; Doughtie, Venton Levy (1964), Design of machine members (4th ed.), McGraw-Hill.
- Industrial Press (2012), Machinery's Handbook (29th ed.), ISBN 978-0-8311-2900-2
- Engineers Edge, Gear Design and Engineering Data.
Otras lecturas
- American Gear Manufacturers Association; American National Standards Institute (2005), Gear Nomenclature: Definitions of Terms with Symbols (ANSI/AGMA 1012-F90 ed.), American Gear Manufacturers Association, ISBN 978-1-55589-846-5.
- Buckingham, Earle (1949), Analytical Mechanics of Gears, McGraw-Hill Book Co..
- Coy, John J.; Townsend, Dennis P.; Zaretsky, Erwin V. (1985), Gearing (PDF), NASA Scientific and Technical Information Branch, NASA-RP-1152; AVSCOM Technical Report 84-C-15.
- Kravchenko A.I., Bovda A.M. Gear with magnetic couple. Pat. of Ukraine N. 56700 – Bul. N. 2, 2011 – F16H 49/00.
- Sclater, Neil. (2011). "Gears: devices, drives and mechanisms." Mechanisms and Mechanical Devices Sourcebook. 5th ed. New York: McGraw Hill. pp. 131–174. ISBN 9780071704427. Drawings and designs of various gearings.
- "Wheels That Can't Slip." Popular Science, February 1945, pp. 120–125.
enlaces externos
- Geararium. Museum of gears and toothed wheels - antique and vintage gears, sprockets, ratchets and other gear-related objects.
- Kinematic Models for Design Digital Library (KMODDL) - movies and photos of hundreds of working models at Cornell University
- Short historical account on the application of analytical geometry to the form of gear teeth
- Mathematical Tutorial for Gearing (Relating to Robotics)
- American Gear Manufacturers Association
- Gear Technology, the Journal of Gear Manufacturing