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65-605: Gleason's importance lies in gear manufacturing — especially in building the machine tools that themselves cut the teeth. These gears and these machines are sold to industrial customers in a wide variety of fields, such as companies in the automotive and aerospace industries. The Gleason Works, the machine shop that eventually evolved into the Gleason Corporation, was founded by Irish immigrant William Gleason in 1865 after his previous experience in other machine shops. An important product came in 1874 with Gleason's invention of

130-638: A ( F A ⋅ e A ⊥ ) − b ( F B ⋅ e B ⊥ ) = a F A − b F B , {\displaystyle F_{\theta }=\mathbf {F} _{A}\cdot {\frac {\partial \mathbf {v} _{A}}{\partial {\dot {\theta }}}}-\mathbf {F} _{B}\cdot {\frac {\partial \mathbf {v} _{B}}{\partial {\dot {\theta }}}}=a(\mathbf {F} _{A}\cdot \mathbf {e} _{A}^{\perp })-b(\mathbf {F} _{B}\cdot \mathbf {e} _{B}^{\perp })=aF_{A}-bF_{B},} where F A and F B are components of

195-421: A bevel gear , whose overall shape is like a slice ( frustum ) of a cone whose apex is the meeting point of the two axes. Bevel gears with equal numbers of teeth and shaft axes at 90 degrees are called miter (US) or mitre (UK) gears. Independently of the angle between the axes, the larger of two unequal matching bevel gears may be internal or external, depending the desired relative sense of rotation. If

260-421: 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. Lever A lever is a simple machine consisting of a beam or rigid rod pivoted at

325-529: A transmission or "gearbox" containing a set of gears that can be meshed in multiple configurations. The gearbox lets the operator vary the torque that is applied to the wheels without changing the engine's speed. Gearboxes are used also in many other machines, such as lathes and conveyor belts . In all those cases, terms like "first gear", "high gear", and "reverse gear" refer to the overall torque ratios of different meshing configurations, rather than to specific physical gears. These terms may be applied even when

390-422: A fixed hinge , or fulcrum . A lever is a rigid body capable of rotating on a point on itself. On the basis of the locations of fulcrum, load and effort, the lever is divided into three types . It is one of the six simple machines identified by Renaissance scientists. A lever amplifies an input force to provide a greater output force, which is said to provide leverage , which is mechanical advantage gained in

455-611: A geared astrolabe was built in Isfahan showing the position of the moon in the zodiac and its phase , and the number of days since new moon. The worm gear was invented in the Indian subcontinent , for use in roller cotton gins , some time during the 13th–14th centuries. A complex astronomical clock, called the Astrarium , was built between 1348 and 1364 by Giovanni Dondi dell'Orologio . It had seven faces and 107 moving parts; it showed

520-419: A great variety of shapes and materials, and are used for many different functions and applications. Diameters may range from a few μm in micromachines , to a few mm in watches and toys to over 10 metres in some mining equipment. Other types of parts that are somewhat similar in shape and function to gears include the sprocket , which is meant to engage with a link chain instead of another gear, and

585-415: A lever is the ratio of output force to input force. M A = F 2 F 1 = a b . {\displaystyle MA={\frac {F_{2}}{F_{1}}}={\frac {a}{b}}.\!} This relationship shows that the mechanical advantage can be computed from ratio of the distances from the fulcrum to where the input and output forces are applied to the lever, assuming

650-517: A motor communicates motion' is from 1814; specifically of a vehicle (bicycle, automobile, etc.) by 1888. A cog is a tooth on a wheel. From Middle English cogge, from Old Norse (compare Norwegian kugg ('cog'), Swedish kugg , kugge ('cog, tooth')), from Proto-Germanic * kuggō (compare Dutch kogge (' cogboat '), German Kock ), from Proto-Indo-European * gugā ('hump, ball') (compare Lithuanian gugà ('pommel, hump, hill'), from PIE * gēw- ('to bend, arch'). First used c. 1300 in

715-682: A pointer on top of the chariot kept the direction of latter unchanged as the chariot turned. Another early surviving example of geared mechanism is a complex calendrical device showing the phase of the Moon, the day of the month and the places of the Sun and the Moon in the Zodiac was invented in the Byzantine empire in the early 6th century AD. Geared mechanical water clocks were built in China by 725 AD. Around 1221 AD,

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780-423: A rigid bar connected to a ground frame by a hinged joint called a fulcrum. The lever is operated by applying an input force F A at a point A located by the coordinate vector r A on the bar. The lever then exerts an output force F B at the point B located by r B . The rotation of the lever about the fulcrum P is defined by the rotation angle θ in radians. Let the coordinate vector of

845-414: A series of teeth that engage with compatible teeth of another gear or other part. The teeth can be integral saliences or cavities machined on the part, or separate pegs inserted into it. In the latter case, the gear is usually called a cogwheel . A cog may be one of those pegs or the whole gear. Two or more meshing gears are called a gear train . The smaller member of a pair of meshing gears

910-605: A series of wooden pegs or cogs around the rim of a wheel. The cogs were often made of maple wood. Wooden gears have been gradually replaced by ones made or metal, such as cast iron at first, then steel and aluminum . Steel is most commonly used because of its high strength-to-weight ratio and low cost. Aluminum is not as strong as steel for the same geometry, but is lighter and easier to machine. powder metallurgy may be used with alloys that cannot be easily cast or machined. Still, because of cost or other considerations, some early metal gears had wooden cogs, each tooth forming

975-400: A type of specialised 'through' mortise and tenon joint More recently engineering plastics and composite materials have been replacing metals in many applications, especially those with moderate speed and torque. They are not as strong as steel, but are cheaper, can be mass-manufactured by injection molding don't need lubrication. Plastic gears may even be intentionally designed to be

1040-415: A weightless lever and no losses due to friction, flexibility or wear. This remains true even though the "horizontal" distance (perpendicular to the pull of gravity) of both a and b change (diminish) as the lever changes to any position away from the horizontal. Levers are classified by the relative positions of the fulcrum, effort and resistance (or load). It is common to call the input force "effort" and

1105-428: Is given by: M A = F B F A = a b . {\displaystyle MA={\frac {F_{B}}{F_{A}}}={\frac {a}{b}}.} This is the law of the lever , which was proven by Archimedes using geometric reasoning. It shows that if the distance a from the fulcrum to where the input force is applied (point A ) is greater than the distance b from fulcrum to where

1170-414: Is no friction in the hinge or bending in the beam. In this case, the power into the lever equals the power out, and the ratio of output to input force is given by the ratio of the distances from the fulcrum to the points of application of these forces. This is known as the law of the lever . The mechanical advantage of a lever can be determined by considering the balance of moments or torque , T , about

1235-425: Is not only acceptable but desirable. For basic analysis purposes, each gear can be idealized as a perfectly rigid body that, in normal operation, turns around a rotation axis that is fixed in space, without sliding along it. Thus, each point of the gear can move only along a circle that is perpendicular to its axis and centered on it. At any moment t , all points of the gear will be rotating around that axis with

1300-440: Is often called pinion . Most commonly, gears and gear trains can be used to trade torque for rotational speed between two axles or other rotating parts and/or to change the axis of rotation and/or to invert the sense of rotation. A gear may also be used to transmit linear force and/or linear motion to a rack , a straight bar with a row of compatible teeth. Gears are among the most common mechanical parts. They come in

1365-512: Is produced by net shape molding. Molded gearing is usually powder metallurgy, plastic injection, or metal die casting. Gears produced by powder metallurgy often require a sintering step after they are removed from the mold. Cast gears require gear cutting or other machining to shape the teeth to the necessary precision. The most common form of gear cutting is hobbing , but gear shaping , milling , and broaching may be used instead. Metal gears intended for heavy duty operation, such as in

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1430-671: Is reversed when one gear wheel drives another gear wheel. Philon of Byzantium was one of the first who used gears in water raising devices. Gears appear in works connected to Hero of Alexandria , in Roman Egypt circa AD 50, but can be traced back to the mechanics of the Library of Alexandria in 3rd-century BC Ptolemaic Egypt , and were greatly developed by the Greek polymath Archimedes (287–212 BC). The earliest surviving gears in Europe were found in

1495-598: Is the Proto-Indo-European stem legwh- , meaning "light", "easy" or "nimble", among other things. The PIE stem also gave rise to the English word "light". The earliest evidence of the lever mechanism dates back to the ancient Near East c.  5000 BC , when it was first used in a simple balance scale . In ancient Egypt c.  4400 BC , a foot pedal was used for the earliest horizontal frame loom . In Mesopotamia (modern Iraq) c.  3000 BC ,

1560-481: Is the generalized coordinate that defines the configuration of the lever, and the generalized force associated with this coordinate is given by F θ = F A ⋅ ∂ v A ∂ θ ˙ − F B ⋅ ∂ v B ∂ θ ˙ =

1625-402: Is the output force. The distances a and b are the perpendicular distances between the forces and the fulcrum. Since the moments of torque must be balanced, T 1 = T 2 {\displaystyle T_{1}=T_{2}\!} . So, F 1 a = F 2 b {\displaystyle F_{1}a=F_{2}b\!} . The mechanical advantage of

1690-456: The Antikythera mechanism an example of a very early and intricate geared device, designed to calculate astronomical positions of the sun, moon, and planets, and predict eclipses . Its time of construction is now estimated between 150 and 100 BC. The Chinese engineer Ma Jun (c. 200–265 AD) described a south-pointing chariot . A set of differential gears connected to the wheels and to

1755-422: The eardrum to the oval window of the cochlea . The lever is a movable bar that pivots on a fulcrum attached to a fixed point. The lever operates by applying forces at different distances from the fulcrum, or a pivot. As the lever rotates around the fulcrum, points further from this pivot move faster than points closer to the pivot. Therefore, a force applied to a point further from the pivot must be less than

1820-443: The shadouf , a crane-like device that uses a lever mechanism, was invented. In ancient Egypt , workmen used the lever to move and uplift obelisks weighing more than 100 tons. This is evident from the recesses in the large blocks and the handling bosses which could not be used for any purpose other than for levers. The earliest remaining writings regarding levers date from the 3rd century BC and were provided, by common belief, by

1885-480: The timing pulley , meant to engage a timing belt . Most gears are round and have equal teeth, designed to operate as smoothly as possible; but there are several applications for non-circular gears , and the Geneva drive has an extremely uneven operation, by design. Gears can be seen as instances of the basic lever "machine". When a small gear drives a larger one, the mechanical advantage of this ideal lever causes

1950-431: The transmissions of cars and trucks, the teeth are heat treated to make them hard and more wear resistant while leaving the core soft but tough . For large gears that are prone to warp, a quench press is used. Gears can be made by 3D printing ; however, this alternative is typically used only for prototypes or very limited production quantities, because of its high cost, low accuracy, and relatively low strength of

2015-451: The 3rd class lever. A compound lever comprises several levers acting in series: the resistance from one lever in a system of levers acts as effort for the next, and thus the applied force is transferred from one lever to the next. Examples of compound levers include scales, nail clippers and piano keys. The malleus , incus and stapes are small bones in the middle ear , connected as compound levers, that transfer sound waves from

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2080-463: The Greek mathematician Archimedes , who famously stated "Give me a lever long enough and a fulcrum on which to place it, and I shall move the world." Autumn Stanley argues that the digging stick can be considered the first lever, which would position prehistoric women as the inventors of lever technology. A lever is a beam connected to ground by a hinge, or pivot, called a fulcrum. The ideal lever does not dissipate or store energy, which means there

2145-559: The Thompson Manufacturing Company of Lancaster, New Hampshire still had a very active business in supplying tens of thousands of maple gear teeth per year, mostly for use in paper mills and grist mills , some dating back over 100 years. The most common techniques for gear manufacturing are dies , sand , and investment casting ; injection molding ; powder metallurgy ; blanking ; and gear cutting . As of 2014, an estimated 80% of all gearing produced worldwide

2210-450: The axes, each section of one gear will interact only with the corresponding section of the other gear. Thus the three-dimensional gear train can be understood as a stack of gears that are flat and infinitesimally thin — that is, essentially two-dimensional. In a crossed arrangement, the axes of rotation of the two gears are not parallel but cross at an arbitrary angle except zero or 180 degrees. For best operation, each wheel then must be

2275-438: The axis, meaning that it is congruent with itself when the gear rotates by 1/ N of a turn. If the gear is meant to transmit or receive torque with a definite sense only (clockwise or counterclockwise with respect to some reference viewpoint), the action surface consists of N separate patches, the tooth faces ; which have the same shape and are positioned in the same way relative to the axis, spaced 1/ N turn apart. If

2340-448: The best shape for each pitch surface is neither cylindrical nor conical but a portion of a hyperboloid of revolution. Such gears are called hypoid for short. Hypoid gears are most commonly found with shafts at 90 degrees. 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

2405-450: The first bevel gear planer, a planer with integral indexing head designed to specialize in planing bevel gears . Planers and indexing heads had been combined before, but never in the winning form factor that Gleason created specifically for gears. Gleason's daughter Kate was integral to the company's early operations, along with brothers James and Andrew. After engineers at Packard developed spiral bevel gears , Gleason pioneered

2470-421: The force located at a point closer in, because power is the product of force and velocity. If a and b are distances from the fulcrum to points A and B and the force F A applied to A is the input and the force F B applied at B is the output, the ratio of the velocities of points A and B is given by a/b , so we have the ratio of the output force to the input force, or mechanical advantage,

2535-400: The forces that are perpendicular to the radial segments PA and PB . The principle of virtual work states that at equilibrium the generalized force is zero, that is F θ = a F A − b F B = 0. {\displaystyle F_{\theta }=aF_{A}-bF_{B}=0.\,\!} Thus, the ratio of the output force F B to

2600-542: The fulcrum to the input point A and to the output point B , respectively. Now introduce the unit vectors e A and e B from the fulcrum to the point A and B , so r A − r P = a e A , r B − r P = b e B . {\displaystyle \mathbf {r} _{A}-\mathbf {r} _{P}=a\mathbf {e} _{A},\quad \mathbf {r} _{B}-\mathbf {r} _{P}=b\mathbf {e} _{B}.} The velocity of

2665-399: The fulcrum. If the distance traveled is greater, then the output force is lessened. T 1 = F 1 a , T 2 = F 2 b {\displaystyle {\begin{aligned}T_{1}&=F_{1}a,\quad \\T_{2}&=F_{2}b\!\end{aligned}}} where F 1 is the input force to the lever and F 2

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2730-426: The input force F A is obtained as M A = F B F A = a b , {\displaystyle MA={\frac {F_{B}}{F_{A}}}={\frac {a}{b}},} which is the mechanical advantage of the lever. This equation shows that if the distance a from the fulcrum to the point A where the input force is applied is greater than the distance b from fulcrum to

2795-506: The machine tools to mass-produce them (with automotive differentials being the primary market). Packard and Gleason settled an infringement lawsuit regarding Packard's patents ( U.S. patent 725,328 and U.S. patent 1,209,682 ) and Gleason's patent ( U.S. patent 1,325,784 ). In 1927, Gleason Works again led the innovation in this market, as it was the first machine tool builder to create machine tools to cut hypoid gears (which are an advanced variant of spiral bevel gears). Gleason

2860-434: The most common configuration, the axes of rotation of the two gears are parallel, and usually their sizes are such that they contact near a point between the two axes. In this configuration, the two gears turn in opposite senses. Occasionally the axes are parallel but one gear is nested inside the other. In this configuration, both gears turn in the same sense. If the two gears are cut by an imaginary plane perpendicular to

2925-522: The most efficient and compact way of transmitting torque between two non-parallel axes. On the other hand, gears are more expensive to manufacture, may require periodic lubrication, and may have greater mass and rotational inertia than the equivalent pulleys. More importantly, the distance between the axes of matched gears is limited and cannot be changed once they are manufactured. There are also applications where slippage under overload or transients (as occurs with belts, hydraulics, and friction wheels)

2990-445: 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. This style of gear is most common in motor vehicle drive trains, in concert with

3055-512: The nymphs of the planthopper insect Issus coleoptratus . The word gear is probably from Old Norse gørvi (plural gørvar ) 'apparel, gear,' related to gøra , gørva 'to make, construct, build; set in order, prepare,' a common verb in Old Norse, "used in a wide range of situations from writing a book to dressing meat". In this context, the meaning of 'toothed wheel in machinery' first attested 1520s; specific mechanical sense of 'parts by which

3120-409: The output force "load" or "resistance". This allows the identification of three classes of levers by the relative locations of the fulcrum, the resistance and the effort: These cases are described by the mnemonic fre 123 where the f fulcrum is between r and e for the 1st class lever, the r resistance is between f and e for the 2nd class lever, and the e effort is between f and r for

3185-399: The output force is applied (point B ), then the lever amplifies the input force. On the other hand, if the distance a from the fulcrum to the input force is less than the distance b from the fulcrum to the output force, then the lever reduces the input force. The use of velocity in the static analysis of a lever is an application of the principle of virtual work . A lever is modeled as

3250-416: The point P that defines the fulcrum be r P , and introduce the lengths a = | r A − r P | , b = | r B − r P | , {\displaystyle a=|\mathbf {r} _{A}-\mathbf {r} _{P}|,\quad b=|\mathbf {r} _{B}-\mathbf {r} _{P}|,} which are the distances from

3315-562: The points A and B are obtained as v A = θ ˙ a e A ⊥ , v B = θ ˙ b e B ⊥ , {\displaystyle \mathbf {v} _{A}={\dot {\theta }}a\mathbf {e} _{A}^{\perp },\quad \mathbf {v} _{B}={\dot {\theta }}b\mathbf {e} _{B}^{\perp },} where e A and e B are unit vectors perpendicular to e A and e B , respectively. The angle θ

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3380-481: The points p and q are moving along different circles; therefore, the contact cannot last more than one instant, and p will then either slide across the other face, or stop contacting it altogether. On the other hand, at any given moment there is at least one such pair of contact points; usually more than one, even a whole line or surface of contact. Actual gears deviate from this model in many ways: they are not perfectly rigid, their mounting does not ensure that

3445-455: The positions of the sun, the moon and the five planets then known, as well as religious feast days. The Salisbury Cathedral clock , built in 1386, it is the world's oldest still working geared mechanical clock. Differential gears were used by the British clock maker Joseph Williamson in 1720. However, the oldest functioning gears by far were created by Nature, and are seen in the hind legs of

3510-672: The resulting part. Besides gear trains, other alternative methods of transmitting torque between non-coaxial parts include link chains driven by sprockets, friction drives , belts and pulleys , hydraulic couplings , and timing belts . One major advantage of gears is that their rigid body and the snug interlocking of the teeth ensure precise tracking of the rotation across the gear train, limited only by backlash and other mechanical defects. For this reason they are favored in precision applications such as watches. Gear trains also can have fewer separate parts (only two) and have minimal power loss, minimal wear, and long life. Gears are also often

3575-425: The rotation axis will be perfectly fixed in space, the teeth may have slightly different shapes and spacing, the tooth faces are not perfectly smooth, and so on. Yet, these deviations from the ideal model can be ignored for a basic analysis of the operation of a gear set. One criterion for classifying gears is the relative position and direction of the axes or rotation of the gears that are to be meshed together. In

3640-456: The same angular speed ω ( t ), in the same sense. The speed need not be constant over time. The action surface of the gear consists of all points of its surface that, in normal operation, may contact the matching gear with positive pressure . All other parts of the surface are irrelevant (except that they cannot be crossed by any part of the matching gear). In a gear with N teeth, the working surface has N -fold rotational symmetry about

3705-524: The sense of 'a wheel having teeth or cogs; late 14c., 'tooth on a wheel'; cog-wheel, early 15c. The gears of the Antikythera mechanism are made of bronze , and the earliest surviving Chinese gears are made of iron, These metals, as well as tin , have been generally used for clocks and similar mechanisms to this day. Historically, large gears, such as used in flour mills , were commonly made of wood rather than metal. They were cogwheels, made by inserting

3770-463: The system, equal to the ratio of the output force to the input force. As such, the lever is a mechanical advantage device , trading off force against movement. The word "lever" entered English around AD 1300 from Old French : levier . This sprang from the stem of the verb lever , meaning "to raise". The verb, in turn, goes back to Latin : levare , itself from the adjective levis , meaning "light" (as in "not heavy"). The word's primary origin

3835-490: The torque T to increase but the rotational speed ω to decrease. The opposite effect is obtained when a large gear drives a small one. The changes are proportional to the gear ratio r , the ratio of the tooth counts. namely, T 2 / T 1 = r = N 2 / N 1 , and ω 2 / ω 1 = 1/ r = N 1 / N 2 . Depending on the geometry of the pair, the sense of rotation may also be inverted (from clockwise to anti-clockwise , or vice-versa). Most vehicles have

3900-416: The torque on each gear may have both senses, the action surface will have two sets of N tooth faces; each set will be effective only while the torque has one specific sense, and the two sets can be analyzed independently of the other. However, in this case the gear usually has also "flip over" symmetry, so that the two sets of tooth faces are congruent after the gear is flipped. This arrangement ensures that

3965-416: The two gears are firmly locked together, at all times, with no backlash . During operation, each point p of each tooth face will at some moment contact a tooth face of the matching gear at some point q of one of its tooth faces. At that moment and at those points, the two faces must have the same perpendicular direction but opposite orientation. But since the two gears are rotating around different axes,

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4030-533: The two gears are sliced by an imaginary sphere whose center is the point where the two axes cross, each section will remain on the surface of that sphere as the gear rotates, and the section of one gear will interact only with the corresponding section of the other gear. In this way, a pair of meshed 3D gears can be understood as a stack of nested infinitely thin cup-like gears. The gears in a matching pair are said to be skew if their axes of rotation are skew lines -- neither parallel nor intersecting. In this case,

4095-517: The vehicle does not actually contain gears, as in a continuously variable transmission . The earliest surviving gears date from the 4th century BC in China (Zhan Guo times – Late East Zhou dynasty ), which have been preserved at the Luoyang Museum of Henan Province, China . In Europe, Aristotle mentions gears around 330 BC, as wheel drives in windlasses. He observed that the direction of rotation

4160-467: The weakest part in a mechanism, so that in case of jamming they will fail first and thus avoid damage to more expensive parts. Such sacrificial gears may be a simpler alternative to other overload-protection devices such as clutches and torque- or current-limited motors. In spite of the advantages of metal and plastic, wood continued to be used for large gears until a couple of centuries ago, because of cost, weight, tradition, or other considerations. In 1967

4225-618: Was a publicly traded company under the symbol 'GLE' on the New York Stock Exchange . In December 1999, it agreed to be acquired by its chairman and chief executive, the senior management, the Gleason Foundation, and private equity firm Vestar Capital Partners . There are multiple companies under the Gleason Corporation umbrella: Gear A gear or gearwheel is a rotating machine part typically used to transmit rotational motion and/or torque by means of

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