Editing Gear (section)

==Tooth profile==

[[File:Tooth surface.jpg|thumb|right|Profile of a spur gear]]
Another criterion to classify gears is the ''tooth profile'', the shape of the cross-section of a tooth face by an imaginary cut perpendicular to the pitch surface, such as the transverse, normal, or axial plane. 

The tooth profile is crucial for the smoothness and uniformity of the movement of matching gears, as well as for the [[friction]] and wear.

=== Artisanal ===

[[File:Storckensohn gears and millstone.jpg|thumb|upright|Wooden cogs set in bevel mortise wheels driving a [[millstone]]. Note wooden spur gears in the background.]]
The teeth of antique or artisanal gears that were cut by hand from sheet material, like those in the Antikhytera mechanism, generally had simple profiles, such as triangles.
<ref name=free2012>{{cite journal |first1=Tony |last1=Freeth |first2=Alexander |last2=Jones |date=February 2012 |title=The Cosmos in the Antikythera Mechanism|url=https://dlib.nyu.edu/awdl/isaw/isaw-papers/4/ |journal=ISAW Papers |number=4 |publisher=[[Institute for the Study of the Ancient World]] |via=[[New York University]] }}</ref> 
The teeth of larger gears —  such as used in windmills — were usually pegs with simple shapes like  cylinders, [[parallelepiped]]s, or triangular [[Prism (geometry)|prisms]] inserted into a smooth wooden or metal wheel; or were holes with equally simple shapes cut into such a wheel.  

Because of their sub-optimal profile, the effective gear ratio of such artisanal matching gears was not constant, but fluctuated over each tooth cycle, resulting in vibrations, noise, and accelerated wear.

=== Cage ===
[[File:Cage Gear.png|thumb|left|300px|Cage gear in Pantigo Windmill, Long Island (with the driving gearwheel disengaged)]]

A ''cage gear'', also called a ''lantern gear'' or ''lantern pinion'', is one of those artisanal  gears having 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|date=October 2013}} 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, a 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. {{cn|date=July 2024}}

=== Mathematical ===

In most modern gears, the tooth profile is usually not straight or circular, but of special form designed to achieve a constant angular velocity ratio.  

There is an infinite variety of tooth profiles that will achieve this goal. In fact, given a fairly arbitrary{{Clarification|reason="what are the constraints?"|date=January 2025}} tooth shape, it is possible to develop a tooth profile for the mating gear that will do it. 

==== Parallel and crossed axes ====

However, two constant velocity tooth profiles are the most commonly used in modern times for gears with parallel or crossed axes, based on the ''[[cycloid gear|cycloid]]'' and ''[[involute gear|involute]]'' curves. 

Cycloidal gears were 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 commonly used in mechanical clocks.

==== Skew axes ====

[[File:Gear-kegelzahnrad.svg|thumb|left|Spiral bevel gears]]
For non-parallel axes with non-straight tooth cuts, the best tooth profile is one of several [[spiral bevel gear]] shapes.  These include 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.<ref name="straightbevel">{{harvnb|McGraw-Hill|2007|p=742}}.</ref> 

The tooth faces in these gear types are not involute cylinders or cones but patches of [[octoidal surface]]s.<ref name=figl2005>{{cite journal | last=Figliolini | first=Giorgio | last2=Angeles | first2=Jorge | title=Algorithms for Involute and Octoidal Bevel-Gear Generation | journal=Journal of Mechanical Design | volume=127 | issue=4 | date=2005-07-01 | issn=1050-0472 | doi=10.1115/1.1900147 | pages=664–672}}</ref> Manufacturing such tooth faces may require a 5-axis [[milling machine]].

Spiral bevel gears have the same advantages and disadvantages relative to their straight-cut cousins as helical gears do to spur gears, such as lower noise and vibration.<ref name="straightbevel">{{harvnb|McGraw-Hill|2007|p=742}}.</ref>   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.<ref>Diss. Hünecke, TU Dresden</ref>

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