Clockwork refers to the inner workings of either mechanical machines called clocks or other mechanisms that work using a complex series of gears. A clockwork mechanism is powered by a clockwork motor consisting of a mainspring, a spiral torsion spring of metal ribbon. Energy is stored in the mainspring manually by winding it up, turning a key attached to a ratchet which twists the mainspring tighter; the force of the mainspring turns the clockwork gears, until the stored energy is used up. The adjectives wind-up and spring-powered refer to mainspring-powered clockwork devices, which include clocks and watches, kitchen timers, music boxes, wind-up toys; the earliest known example of a clockwork mechanism is the Antikythera mechanism, a first-century BC geared analogue computer, somewhat astrolabe-like, for calculating astronomical positions and eclipses, recovered from a Greek shipwreck. There are many other accounts of clockwork devices in ancient Greece in its mythology, the mechanism itself is sophisticated enough to indicate a significant history of lesser devices leading up to its creation.
At some point, this level of sophistication in clockwork technology was lost or forgotten in Europe, only returned when brought from the Islamic world after the Crusades, along with other knowledge leading to the Renaissance. Clockwork recovered the equivalent of pre-Roman technological levels in the 14th century; as in Greek mythology, there are ambitious automation claims in the legends of other cultures. For example, in Jewish legend, Solomon used his wisdom to design a throne with mechanical animals which hailed him as king when he ascended it. It's said that when King Solomon stepped upon the throne, a mechanism was set in motion; as soon as he stepped upon the first step, a golden ox and a golden lion each stretched out one foot to support him and help him rise to the next step. On each side, the animals helped the King up. In ancient China, a curious account of automation is found in the Lie Zi text, written in the 3rd century BC. Within it there is a description of a much earlier encounter between King Mu of Zhou and a mechanical engineer known as Yan Shi, an'artificer'.
The latter proudly presented the king with a life-size, human-shaped figure of his mechanical handiwork: The king stared at the figure in astonishment. It walked with rapid strides, moving its head up and down, so that anyone would have taken it for a live human being; the artificer touched its chin, it began singing in tune. He touched its hand, it began posturing, keeping perfect time... As the performance was drawing to an end, the robot winked its eye and made advances to the ladies in attendance, whereupon the king became incensed and would have had Yen Shih executed on the spot had not the latter, in mortal fear taken the robot to pieces to let him see what it was. And, indeed, it turned out to be only a construction of leather, wood and lacquer, variously coloured white, black and blue. Examining it the king found all the internal organs complete—liver, heart, spleen, kidneys and intestines; the king tried the effect of taking away the heart, found that the mouth could no longer speak.
The king was delighted. Other notable examples include Archytas's dove, mentioned by Aulus Gellius. Similar Chinese accounts of flying automata are written of the 5th century BC Mohist philosopher Mozi and his contemporary Lu Ban, who made artificial wooden birds that could fly, according to the Han Fei Zi and other texts. By the 11th century, clockwork was used for both timepieces and to track astronomical events, in Europe; the clocks did not keep time accurately by modern standards, but the astronomical devices were used to predict the positions of planets and other movement. The same timeline seems to apply in Europe, where mechanical escapements were used in clocks by that time. Up to the 15th century, clockwork was driven by water, weights, or other roundabout primitive means, but in 1430 a clock was presented to Philip the Good, Duke of Burgundy, driven by a spring; this became a standard technology along with weight-driven movements. In the mid-16th century, Christiaan Huygens took an idea from Galileo Galilei and developed it into the first modern pendulum mechanism.
However, whereas the spring or the weight provided the motive power, the pendulum controlled the rate of release of that power via some escape mechanism at a regulated rate. The Smithsonian Institution has in its collection a clockwork monk, about 15 in high dating as early as 1560; the monk is driven by a key-wound spring and walks the path of a square, striking his chest with his right arm, while raising and lowering a small wooden cross and rosary in his left hand and nodding his head, rolling his eyes, mouthing silent obsequies. From time to time, he brings the cross to his kisses it, it is believed that the monk was manufactured by Juanelo Turriano, mechanician to the Holy Roman Emperor Charles V. Power for the device is stored within it, via a winding device that applies mechanical stress to an energy-storage mechanism such as a mainspring, thus involving some form of escapement; the use
Swiss made is a label or marker used to indicate that a product was made or assembled in Switzerland or the geographic Swiss region. According to the Federal Act on the Protection of Trade Marks and Indications of Source, a good or service may be designated "Swiss made" if:For food products: 80% of the weight of the raw materials and the essential processing must take place in Switzerland. For industrial products: 60% of the manufacturing costs and the essential manufacturing step must occur in Switzerland. For services: the company headquarters and administration must be located in Switzerland. Most associated with watches or timepieces made in Switzerland, a watch is considered Swiss if its movement is Swiss, if the movement has been assembled in the Swiss region, its final inspection occurred in Switzerland, at least 60% of manufacturing costs are domestic. Besides the Swiss made requirements, watches may carry the Swiss movement marker if at least half of the assembled parts are of Swiss manufacture.
In addition to "Swiss made" and "Swiss Movt", under Swiss law watches may carry the words "Suisse", "produit suisse", "fabriqué en Suisse", "qualité suisse" or the English translation, "Swiss". The wording was formally adopted in the late 19th century and is unique in that most other countries use the phrase "Made in"; the most obvious place where the label is found is on Swiss watches. The Swiss laws permit the use of the words "Suisse", "produit suisse", "fabriqué en Suisse", "qualité suisse" or the translations, "Swiss", "Swiss made", or "Swiss Movement". On some older watches, for example, the word "Swiss" appears alone on the dial at the six o'clock position. There are two discrete sections of the Swiss law; the first law, which applies to all types of Swiss products, is the "Federal Act on the Protection of Trade Marks and Indications of Source". Its article 50 provided the authority for the enactment of the second law, Ordonnance du 23 décembre 1971, relating Swiss watches; the text of either law is available in French, German or Italian, since those are the principal official languages of Switzerland.
The aforementioned Swiss legal standards permit watch brands or watchmakers to label watches Swiss made under certain defined circumstances. These standards have changed over time and were not always codified in the national law, so older watches which bear the mark Swiss made may not meet the current legal definition. On the other hand, they might well exceed the current legal definition of Swiss made. Indeed, the current law of the applicability of Swiss made was codified on December 23, 1971; the Ordinance regulating the use of the name "Swiss" on watches first defines a "watch" by the dimensions of its movement. Thereafter, the law defines a Swiss watch, the definition of, dependent on certain aspects of its movement; the law goes on to define under what circumstances a watch movement may be considered Swiss made. The law sets forth the conditions for the use of the name Swiss on watches, on watch cases, on watch movements, on watch dials and on replacement watch parts. In sum, a watch is considered Swiss whose movement is encased in Switzerland and whose final control by the Manufacture d'horlogerie takes place in Switzerland.
The legal standards for the use of "Swiss made" on a watch are a minimum standard, the Swissness of a watch is dependent on the brand and its reputation. The Swiss watch industry has not reached an agreement over the specific definition of Swiss made, as some companies favor stricter regulations and others prefer including lower-cost foreign components; the Swiss Federal Council modified the ordinance regulating the use of the "Swiss" name for watches in 1995. This revision was explained in a press release entitled On foreign parts for watches. A watch is considered Swiss, according to the Swiss law if: its movement is Swiss and, its movement is cased up in Switzerland and. From 1 January 2017, the law set the minimum at 60 percent. If a watch movement is intended for export and will not be cased-up in Switzerland, but it otherwise meets the criteria to be considered a Swiss movement, the watch may say "Swiss Movement" but it may not say Swiss made on the watch case or dial. A watch that says "Swiss Quartz" is considered to be a proper Swiss watch.
However, it is improperly used by foreign manufacturers to indicate that the quartz movement is of Swiss origin. Use of the Swiss made label for watches is covered by an ordinance of the Federal Council dated 29 December 1971; the Swiss standard is pejoratively referred to as the 60% Rule. However, it has its basis in real life economics. Again, the law sets forth a minimum standard; the famous or infamous Swiss Made Ordinance has, for a number of years, been subject to many criticisms inside the industry, because it is considered too lax, but in legal circles, where the view is that it no longer meets the legal mandate specified in the companion law on trademarks. It is not known that quite a few Swiss companies have watches assembled in China for export to North America and Europe, where the brand name is more important than the “Swiss made” label; such watches may consist of a Chinese case and a Chinese crystal, a Taiwan-made dial and metal bracelet and Japanese
A mainspring is a spiral torsion spring of metal ribbon—commonly spring steel—used as a power source in mechanical watches, some clocks, other clockwork mechanisms. Winding the timepiece, by turning a knob or key, stores energy in the mainspring by twisting the spiral tighter; the force of the mainspring turns the clock's wheels as it unwinds, until the next winding is needed. The adjectives wind-up and spring-powered refer to mechanisms powered by mainsprings, which include kitchen timers, music boxes, wind-up toys and clockwork radios. A modern watch mainspring is a long strip of hardened and blued steel, or specialised steel alloy, 20–30 cm long and 0.05-0.2 mm thick. The mainspring in the common 1-day movement is calculated to enable the watch to run for 36 to 40 hours, i.e. 24 hours between daily windings with a power-reserve of 12 to 16 hours, in case the owner is late winding the watch. This is the normal standard for hand-wound as well as self-winding watches. 8-Day movements, used in clocks meant to be wound weekly, provide power for at least 192 hours but use longer mainsprings and bigger barrels.
Clock mainsprings are similar to watch springs, only larger. Since 1945, carbon steel alloys have been superseded by newer special alloys, by cold-rolled alloys. Known to watchmakers as ` white metal' springs, these have a higher elastic limit, they are less subject to permanent bending and there is scarcely any risk of their breaking. Some of them are practically non-magnetic. In their relaxed form, mainsprings are made in three distinct shapes: Spiral coiled: These are coiled in the same direction throughout, in a simple spiral. Semi-reverse: The outer end of the spring is coiled in the reverse direction for less than one turn. Reverse: the outer end of the spring is coiled in the reverse direction for one or more turns; the semi-reverse and reverse types provide extra force at the end of the running period, when the spring is out of energy, in order to keep the timepiece running at a constant rate to the end. The mainspring is coiled around an axle called the arbor, with the inner end hooked to it.
In many clocks, the outer end is attached to a stationary post. The spring is wound up by turning the arbor, after winding its force turns the arbor the other way to run the clock; the disadvantage of this open spring arrangement is that while the mainspring is being wound, its drive force is removed from the clock movement, so the clock may stop. This type is used on alarm clocks, music boxes and kitchen timers where it doesn't matter if the mechanism stops while winding; the winding mechanism always has a ratchet attached, with a pawl to prevent the spring from unwinding. In the form used in modern watches, called the going barrel, the mainspring is coiled around an arbor and enclosed inside a cylindrical box called the barrel, free to turn; the spring is attached to the arbor at its inner end, to the barrel at its outer end. The attachments are small hooks or tabs, which the spring is hooked to by square holes in its ends, so it can be replaced; the mainspring drives the watch movement by the barrel.
Winding the watch turns the arbor, which tightens the mainspring, wrapping it closer around the arbor. The arbor has a ratchet attached to it, with a click to prevent the spring from turning the arbor backward and unwinding. After winding, the arbor is stationary and the pull of the mainspring turns the barrel, which has a ring of gear teeth around it; this meshes with one of the clocks gears the center wheel pinion and drives the wheel train. The barrel rotates once every 8 hours, so the common 40-hour spring requires 5 turns to unwind completely; the mainspring contains a lot of energy. Clocks and watches have to be disassembled periodically for maintenance and repair, if precautions are not taken the spring can release causing serious injury. Mainsprings are'let down' before servicing, by pulling the click back while holding the winding key, allowing the spring to unwind; however in their'let down' state, mainsprings contain dangerous residual tension. Watchmakers and clockmakers use a tool called a "mainspring winder" to safely install and remove them.
Large mainsprings in clocks are immobilized by "mainspring clamps" before removal. Mainsprings appeared in 15th-century Europe, it replaced the weight hanging from a cord wrapped around a pulley, the power source used in all previous mechanical clocks. Around 1400 coiled springs began to be used in locks, many early clockmakers were locksmiths. Springs were applied to clocks to make them smaller and more portable than previous weight-driven clocks, evolving into the first pocketwatches by 1600. Many sources erroneously credit the invention of the mainspring to the Nuremberg clockmaker Peter Henlein around 1511. However, many references in 15th-century sources to portable clocks'without weights', at least two surviving examples, show that spring-driven clocks existed by the early years of that century; the oldest surviving clock powered by a mainspring is the Burgunderuhr, an ornate, gilt chamber clock at the Germanisches Nationalmuseum in Nuremberg, whose iconography suggests that it was made around 1430 for Philip the Good, Duke of Burgundy.
The first mainsprings were made of steel without tempering or hardening pro
The Spring Drive is a watch movement conceived in 1977 by Yoshikazu Akahane at Suwa Seikosha. Commercially released in 1999, the movement is found in watches distributed by the Seiko Watch Corporation subsidiary of Seiko Holdings, including its Credor, Grand Seiko, Prospex brands. Specified to one second accuracy per day, the movement uses a conventional power train as in traditional mechanical watches, but rather than an escapement or balance wheel, instead features Seiko's Tri-synchro Regulator system in which a reference quartz signal is used along with a magnetic brake to regulate the power delivery to the watch hands. One of its unique features is a true continuously sweeping second hand, instead of the traditional beats per time unit as seen with traditional mechanical and most quartz watches. Most quartz watches are electronically regulated to a single "tick" per second to conserve battery life; the original Bulova Accutron tuning fork models featured a near continuous moving seconds hand whereas their contemporary ultra-high frequency and Precisionist quartz movements are regulated at high rates to give the illusion of continuous movement.
Spring Drive uses a mainspring, automatic winder and stem winding like in a mechanical watch to store the watch energy. The conventional escapement is replaced with a device that Seiko calls a Tri-synchro Regulator to regulate the unwinding of the mainspring; the regulator controls the use of the three forms of energy used in the Spring Drive mechanism. The energy produced by the glide wheel is used to power a control circuit and quartz crystal oscillator, which in turn regulates the electro-mechanical braking of the glide wheel; the glide wheel's speed is sampled 8 times per second and compared with the reference quartz signal by the circuit. A variable braking force is continuously applied to regulate the glide wheel's frequency; this is the only movement with a time-only feedback loop in existence today. The Tri-synchro Regulator's innovations result in a watch where the hands glide instead of ticking as in a conventional mechanical or quartz watch; this is because the movement never stops as in a traditional escapement, it is slowed to the proper speed by the brake.
The movement is specified to 1 second accuracy per day. However, 1–2s per week is reported by owners; the movement is used on the Spring Drive International collection and in some watches of Grand Seiko, Galante, Izul and Prospex series. Complications include moon phase, power reserve, sonnerie, GMT and calendar functions; these watches are expensive, with the least complicated models costing several thousand dollars. The top of the line is the Credor Minute Repeater in rose gold priced at 33 million yen and can be considered a grand complication; the design was first conceived by Yoshikazu Akahane at Suwa Seikosha in 1977, patents were applied for it in 1982. In total, no less than 230 patents have been applied worldwide for this movement; the movement was announced publicly in 1997, presented at the 1998 Basel Watch Fair and first appeared commercially in the Credor'luxury watch' range as a limited edition in 1999. A version which included an automatic winder was shown in Seiko models at the 2005 Basel Watch Fair.
The internationally available Seiko models were launched at the Musée d'Orsay in Paris on September 14, went on sale the following day. Swatch's research company ASULAB has developed a conceptually similar movement called the High Precision Mechanics movement. Several proof of concept prototypes based on the ETA 2824 caliber were produced in the late 1990s. Seiko's efforts with the Spring Drive predates ASULAB's HPM, since Spring Drive watches were on sale in 1999. Early models, manual wind and 48h power reserve: 7R68: 30 jewels, date. 7R78: 30 jewels, date. 7R88: 30 jewels, date. 7R99: 32 jewels. Current calibers with standard features. Time accuracy: monthly rate within ±15 sec and power reserve indicator. 5R64: 32 jewels, small seconds hand. 5R65: 30 jewels, date. 5R66: 30 jewels, date, GMT. 5R67: 30 jewels, Moon Phase indicator. 5R77: 30 jewels, Moon Phase indicator. 5R85: 49 jewels, Chronograph, Izul. 5R86: 50 jewels, date, GMT, Spacewalk. 7R06: 88 jewels, manual winding, Sonnerie. 7R08: 44 jewels, manual winding, Eichi I.
7R11: 112 jewels, manual winding, Minute Repeater. 7R14: 41 jewels, manual winding, Eichi II. 9R01: 56 jewels, manual winding. 9R15: 30 jewels, date. 9R65: 30 jewels, date. 9R66: 30 jewels, date, GMT. 9R84: 41 jewels, Chronograph. 9R86: 50 jewels, date, GMT, Chronograph. 9R96: 50 jewels, date, GMT, Chronograph. Grand Seiko Spring Drive Collection - Updated September 2018 Spring Drive site. SPRING DRIVE 9R movement SEIKO Technology: Spring Drive
A watch is a timepiece intended to be carried or worn by a person. It is designed to keep working despite the motions caused by the person's activities. A wristwatch is designed to be worn around the wrist, attached by a watch strap or other type of bracelet. A pocket watch is designed for a person to carry in a pocket; the study of timekeeping is known as horology. Watches progressed in the 17th century from spring-powered clocks, which appeared as early as the 14th century. During most of its history the watch was a mechanical device, driven by clockwork, powered by winding a mainspring, keeping time with an oscillating balance wheel; these are called mechanical watches. In the 1960s the electronic quartz watch was invented, powered by a battery and kept time with a vibrating quartz crystal. By the 1980s the quartz watch had taken over most of the market from the mechanical watch; this is called the quartz revolution. Developments in the 2010s include smartwatches, which are elaborate computer-like electronic devices designed to be worn on a wrist.
They incorporate timekeeping functions, but these are only a small subset of the smartwatch's facilities. In general, modern watches display the day, date and year. For mechanical watches, various extra features called "complications", such as moon-phase displays and the different types of tourbillon, are sometimes included. Most electronic quartz watches, on the other hand, include time-related features such as timers and alarm functions. Furthermore, some modern smartwatches incorporate calculators, GPS and Bluetooth technology or have heart-rate monitoring capabilities, some of them use radio clock technology to correct the time. Today, most watches in the market that are inexpensive and medium-priced, used for timekeeping, have quartz movements. However, expensive collectible watches, valued more for their elaborate craftsmanship, aesthetic appeal and glamorous design than for simple timekeeping have traditional mechanical movements though they are less accurate and more expensive than electronic ones.
As of 2018, the most expensive watch sold at auction is the Patek Philippe Henry Graves Supercomplication, the world's most complicated mechanical watch until 1989, fetching 24 million US dollars in Geneva on November 11, 2014. Watches evolved from portable spring-driven clocks. Watches were not worn in pockets until the 17th century. One account says that the word "watch" came from the Old English word woecce which meant "watchman", because it was used by town watchmen to keep track of their shifts at work. Another says that the term came from 17th century sailors, who used the new mechanisms to time the length of their shipboard watches. A great leap forward in accuracy occurred in 1657 with the addition of the balance spring to the balance wheel, an invention disputed both at the time and since between Robert Hooke and Christiaan Huygens; this innovation increased watches' accuracy enormously, reducing error from several hours per day to 10 minutes per day, resulting in the addition of the minute hand to the face from around 1680 in Britain and 1700 in France.
The increased accuracy of the balance wheel focused attention on errors caused by other parts of the movement, igniting a two-century wave of watchmaking innovation. The first thing to be improved was the escapement; the verge escapement was replaced in quality watches by the cylinder escapement, invented by Thomas Tompion in 1695 and further developed by George Graham in the 1720s. Improvements in manufacturing such as the tooth-cutting machine devised by Robert Hooke allowed some increase in the volume of watch production, although finishing and assembling was still done by hand until well into the 19th century. A major cause of error in balance wheel timepieces, caused by changes in elasticity of the balance spring from temperature changes, was solved by the bimetallic temperature compensated balance wheel invented in 1765 by Pierre Le Roy and improved by Thomas Earnshaw; the lever escapement was the single most important technological breakthrough, was invented by Thomas Mudge in 1759 and improved by Josiah Emery in 1785, although it only came into use from about 1800 onwards, chiefly in Britain.
The British had predominated in watch manufacture for much of the 17th and 18th centuries, but maintained a system of production, geared towards high-quality products for the elite. Although there was an attempt to modernise clock manufacture with mass production techniques and the application of duplicating tools and machinery by the British Watch Company in 1843, it was in the United States that this system took off. Aaron Lufkin Dennison started a factory in 1851 in Massachusetts that used interchangeable parts, by 1861 it was running a successful enterprise incorporated as the Waltham Watch Company; the concept of the wristwatch goes back to the production of the earliest watches in the 16th century. Elizabeth I of England received a wristwatch from Robert Dudley in 1571, described as an armed watch; the oldest surviving wristwatch is one given to Joséphine de Beauharnais. From the beginning, wristwatches were exclusively worn by women, while men used pocket watches up until the early 20th century.
Wristwatches were first worn by military men towards the end of the 19th century, when the importance of synchronizing maneuvers during war, without revealing the plan to the enemy through signaling, was recognized. The Garstin Company of London patented a "Watch Wristlet" design in 1893, but they were producing similar designs from the 1880s
A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position; when released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging back and forth. The time for one complete cycle, a left swing and a right swing, is called the period; the period depends on the length of the pendulum and to a slight degree on the amplitude, the width of the pendulum's swing. From the first scientific investigations of the pendulum around 1602 by Galileo Galilei, the regular motion of pendulums was used for timekeeping, was the world's most accurate timekeeping technology until the 1930s; the pendulum clock invented by Christian Huygens in 1658 became the world's standard timekeeper, used in homes and offices for 270 years, achieved accuracy of about one second per year before it was superseded as a time standard by the quartz clock in the 1930s.
Pendulums are used in scientific instruments such as accelerometers and seismometers. They were used as gravimeters to measure the acceleration of gravity in geophysical surveys, as a standard of length; the word "pendulum" is new Latin, from the Latin pendulus, meaning'hanging'. The simple gravity pendulum is an idealized mathematical model of a pendulum; this is a weight on the end of a massless cord suspended without friction. When given an initial push, it will swing forth at a constant amplitude. Real pendulums are subject to friction and air drag, so the amplitude of their swings declines; the period of swing of a simple gravity pendulum depends on its length, the local strength of gravity, to a small extent on the maximum angle that the pendulum swings away from vertical, θ0, called the amplitude. It is independent of the mass of the bob. If the amplitude is limited to small swings, the period T of a simple pendulum, the time taken for a complete cycle, is: T ≈ 2 π L g θ 0 ≪ 1 r a d i a n where L is the length of the pendulum and g is the local acceleration of gravity.
For small swings the period of swing is the same for different size swings: that is, the period is independent of amplitude. This property, called isochronism, is the reason. Successive swings of the pendulum if changing in amplitude, take the same amount of time. For larger amplitudes, the period increases with amplitude so it is longer than given by equation. For example, at an amplitude of θ0 = 23 ° it is 1 % larger; the period increases asymptotically as θ0 approaches 180°, because the value θ0 = 180° is an unstable equilibrium point for the pendulum. The true period of an ideal simple gravity pendulum can be written in several different forms, one example being the infinite series: T = 2 π L g where θ 0 is in radians; the difference between this true period and the period for small swings above is called the circular error. In the case of a typical grandfather clock whose pendulum has a swing of 6° and thus an amplitude of 3°, the difference between the true period and the small angle approximation amounts to about 15 seconds per day.
For small swings the pendulum approximates a harmonic oscillator, its motion as a function of time, t, is simple harmonic motion: θ = θ 0 cos where φ is a constant value, dependent on initial conditions. For real pendulums, the period varies with factors such as the buoyancy and viscous resistance of the air, the mass of the string or rod, the size and shape of the bob and how it is attached to the string, flexibility and stretching of the string. In precision applications, corrections for these factors may need to be applied to eq. to give the period accurately. Any swinging rigid body free to rotate about a fixed horizontal axis is called a compound pendulum or physical pendulum; the appropriate equivalent length L for calculating the period of any such pendulum is the distance from the pivot to the center of oscillation. This point is located under the center of mass at a distance from the pivot traditionally called the radius of oscillation, which depends on the mass distribution of the pendulum.
If most of the mass is concentrated in a small bob compared to the pendulum length, the center of oscillation is close to the center of mass. The radiu
A gear train is a mechanical system formed by mounting gears on a frame so the teeth of the gears engage. Gear teeth are designed to ensure the pitch circles of engaging gears roll on each other without slipping, providing a smooth transmission of rotation from one gear to the next; the transmission of rotation between contacting toothed wheels can be traced back to the Antikythera mechanism of Greece and the south-pointing chariot of China. Illustrations by the Renaissance scientist Georgius Agricola show gear trains with cylindrical teeth; the implementation of the involute tooth yielded a standard gear design that provides a constant speed ratio. Features of gears and gear trains include: The ratio of the pitch circles of mating gears defines the speed ratio and the mechanical advantage of the gear set. A planetary gear train provides high gear reduction in a compact package, it is possible to design gear teeth for gears that are non-circular, yet still transmit torque smoothly. The speed ratios of chain and belt drives are computed in the same way as gear ratios.
See bicycle gearing. Gear teeth are designed so the number of teeth on a gear is proportional to the radius of its pitch circle, so the pitch circles of meshing gears roll on each other without slipping; the speed ratio for a pair of meshing gears can be computed from ratio of the radii of the pitch circles and the ratio of the number of teeth on each gear. The velocity v of the point of contact on the pitch circles is the same on both gears, is given by v = r A ω A = r B ω B, where input gear A with radius rA and angular velocity ωA meshes with output gear B with radius rB and angular velocity ωB. Therefore, ω A ω B = r B r A = N B N A. where NA is the number of teeth on the input gear and NB is the number of teeth on the output gear. The mechanical advantage of a pair of meshing gears for which the input gear has NA teeth and the output gear has NB teeth is given by M A = N B N A; this shows that if the output gear GB has more teeth than the input gear GA the gear train amplifies the input torque.
And, if the output gear has fewer teeth than the input gear the gear train reduces the input torque. If the output gear of a gear train rotates more than the input gear the gear train is called a speed reducer. In this case, because the output gear must have more teeth than the input gear, the speed reducer amplifies the input torque. For this analysis, we consider a gear train that has one degree-of-freedom, which means the angular rotation of all the gears in the gear train are defined by the angle of the input gear; the size of the gears and the sequence in which they engage define the ratio of the angular velocity ωA of the input gear to the angular velocity ωB of the output gear, known as the speed ratio, or gear ratio, of the gear train. Let R be the speed ratio ω A ω B = R; the input torque TA acting on the input gear GA is transformed by the gear train into the output torque TB exerted by the output gear GB. If we assume the gears are rigid and there are no losses in the engagement of the gear teeth the principle of virtual work can be used to analyze the static equilibrium of the gear train.
Let the angle θ of the input gear be the generalized coordinate of the gear train the speed ratio R of the gear train defines the angular velocity of the output gear in terms of the input gear: ω A = ω, ω B = ω / R. The formula for the generalized force obtained from the principle of virtual work with applied torques yields: F θ = T A ∂ ω A ∂ ω − T B ∂ ω B ∂ ω = T A − T B / R = 0; the mechanical advantage of the gear train is the ratio of the output torque TB to the input torque TA, the above equation yields: M A = T B T A = R. The speed ratio of a gear train defines its mechanical advantage; this shows that if the input gear rotates faster than the output gear the gear train amplifies the input torque. And if the input gear rotates slower than the output gear, the gear train reduces the input torque; the simplest example of a gear train has two gears. The "input gear" transmits power to the "output gear"; the input gear will be connected to a power source, such as a motor or engine. In such a