GEARS: AN HISTORICAL NOTE

Gears have existed since the invention of rotating machinery. Because of their

force-multiplying properties, early engineers used them for hoisting heavy loads such as

building materials. The mechanical advantage of gears was also used for ship anchor

hoists and catapult pre-tensioning.

Early gears were made from wood with cylindrical pegs for cogs and were often

lubricated with animal fat grease. Gears were also used in wind and water wheel

machinery for decreasing or increasing the provided rotational speed for application to

pumps and other powered machines. An early gear arrangement used to power textile

machinery is illustrated in the following figure. The rotational speed of a water or horse

drawn wheel was typically too slow to use, so a set of wooden gears needed to be used

to increase the speed to a usable level.

 

An 18th Century Application of Gears for Powering Textile Machinery

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The industrial revolution in Britain in the eighteenth century saw an explosion in the use

of metal gearing. A science of gear design and manufacture rapidly developed through

the nineteenth century.

Today, the most significant new gear developments are in the area of materials. Modern

metallurgy has greatly increased the useful life of industrial and automotive gears, and

consumer electronics has driven plastic gearing to new levels of lubricant-free reliability

and quiet operation.

GEARA gear or cogwheel is a rotating machine part having cut teeth, or cogs,

which mesh with another toothed part in order to transmit torque, in most cases with

teeth on the one gear being of identical shape, and often also with that shape on the

other gear. Two or more gears working in tandem are called a transmission and can

produce a mechanical advantage through a gear ratio and thus may be considered

a simple machine. Geared devices can change the speed, torque, and direction of

a power source. The most common situation is for a gear to mesh with another gear;

however, a gear can also mesh with a non-rotating toothed part, called a rack, thereby

producing translation instead of rotation.

The gears in a transmission are analogous to the wheels in a crossed

belt pulley system. An advantage of gears is that the teeth of a gear prevent slippage.

When two gears mesh, and one gear is bigger than the other (even though the size of

the teeth must match), a mechanical advantage is produced, with the rotational

speeds and the torques of the two gears differing in an inverse relationship.

In transmissions which offer multiple gear ratios, such as bicycles, motorcycles, and

cars, the term gear, as in first gear, refers to a gear ratio rather than an actual physical

gear. The term is used to describe similar devices even when the gear ratio

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is continuous rather than discrete, or when the device does not actually contain any

gears, as in a continuously variable transmission.[1]

The earliest known reference to gears was circa A.D. 50 by Hero of Alexandria,[2] but

they can be traced back to the Greek mechanics of theAlexandrian school in the 3rd

century B.C. and were greatly developed by the Greek polymath Archimedes (287–212

B.C.).[3] The Antikythera mechanism is an example of a very early and intricate geared

device, designed to calculate astronomical positions. Its time of construction is now

estimated between 150 and 100 BC.[4]

COMPARISON WITH DRIVE MECHANISMS

The definite velocity ratio which results from having teeth gives gears an

advantage over other drives (such as traction drives and V-belts) in precision

machines such as watches that depend upon an exact velocity ratio. In cases

where driver and follower are proximal, gears also have an advantage over other

drives in the reduced number of parts required; the downside is that gears are

more expensive to manufacture and their lubrication requirements may impose a

higher operating cost.

TYPES OF GEARS

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EXTERNAL VS INTERNAL GEARS

An external gear is one with the teeth formed on the outer surface of a cylinder or

cone. Conversely, an internal gear is one with the teeth formed on the inner surface of a

cylinder or cone. For bevel gears, an internal gear is one with the pitch angle exceeding

90 degrees. Internal gears do not cause output shaft direction reversal.

SPUR GEARS

Spur Gears are the most common and also the oldest type of gear

available. They are used in many different machines for a wide variety of functions.

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Although they are very simple in design, several different, discrete types of spur gears

exist. Anyone who is interested in mechanical engineering or who is about to undertake

a project that involves the use of gears of any type ought to familiarize themselves with

spur gears and the ways in which they are used.

TYPES OF SPUR GEARS

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Spur Gears in General

In general, spur gears are round metal disks with teeth cut around the circumference. In

order for the gear to qualify as a spur gear, the teeth must be cut so that they run

perpendicular to the gear's face. Spur gears are the simplest design of gear produced.

They are usually used for the transmission of rotary force. For instance, if two shafts are

parallel to one another, and one is spinning, a spur gear can help to transfer that force

onto the other shaft. Spur gears usually have an operating efficiency of 98% to 99%.

Anti-Backlash Spur Gears

Antibacklash spur gears, as their name suggests, have little to no backlash, and so are

used in high-precision applications. Often, these spur gears are built with springs for

proper tensoring. They are usually built from brass, aluminum or stainless steel. In order

for antibacklash spur gears to work together, they must have the same diametral pitch

and pressure angle.

Cluster Spur Gears

Cluster spur gears come "clustered" together, usually on the same shaft, and have

varying diameters.

Spur Gear Blanks

Spur gear blanks are spur gears with no teeth cut into them. These types of spur gears

can be useful if you do not yet know the precise number of teeth you will require for your

spur gear's application.

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Pinion Shafts

Pinion shafts are basically stretched-out spur gears, cylinders with teeth running for

their entire length.

Ratchets and Pawls

Ratchets and pawls are two spur gears that work together. Ratchets are gear wheels

with teeth, while pawls are spring-loaded and pivot. Pawls are usually slanted.

Together, these gears allow for unidirectional movement.

Clamp Hub Spur Gears

Clamp hub spur gears are named for the manner in which they connect to their shaft,

that is, with a clamp at the spur gear's center.

Hubless Spur Gears

Hubless spur gears have no hub and instead connect to their shafts through friction or

with adhesive.

Pin Hub Spur Gears

Pin hub spur gears connect to their shaft through the use of a removable pin as

opposed to a clamp.

Spur Gear Materials

Spur gears can be built from a wide variety of different materials. Different materials are

often used for different types of applications. For instance, cast iron spur gears are

strong and are used in many commercial applications. Aluminum alloy spur gears,

conversely, are lighter but can also be built into more precise shapes, so they are used

7 | P a g e

for low-impact applications requiring high precision, such as measuring instruments.

Other materials from which spur gears are built include cast steels, carbon steels,

stainless steels, brass, magnesium and titanium alloys, and sometimes even nylon.

MATERIALS USED FOR SPUR GEARS

Acetal

Acetal is a plastic polymer that is used either in its pure state or slightly altered state---

e.g. Derlin---for a number of spur gears. The acetal polymer is much stronger than

common plastic, though it can be easily molded to any shape, including a spur gear.

Once acetal has hardened in the shape of a spur gear, it is stif, strong and resistant to

abrasion. The malleability, strength and resilience make it an ideal material for spur

gears.

Cast Iron

Cast iron is, like acetal, an easily molded material. It is also highly resistant to rust. Cast

iron is not pure iron, and because of this, any given batch of cast iron will have different

ingredients. These different ingredients cohere for different degrees of strength and

durability. Cast iron is used in machine parts because it is relatively inexpensive, rust

resistant and easy to mold, though it may be either incredibly strong or incredibly weak,

depending upon the admixture.

Stainless Steel

Stainless steel is a metal alloy commonly used in the casting of spur gears. A metal

alloy is a metal composed of two or more distinct elements that are melted together.

Like cast iron, it is highly resistant to oxidation, and like acetal, it is resistant to

abrasions and other weakening blemishes. Stainless steels resistance to rust and

scarring is due to the infusion of chromium. The strength, durability and corrosion

resistance make stainless steel a popular material for spur gears.

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SKEW GEARS

For a 'crossed' or 'skew' configuration, the gears must have the same pressure

angle and normal pitch; however, the helix angle and handedness can be different. The

relationship between the two shafts is actually defined by the helix angle(s) of the two

shafts and the handedness, as defined:[9]

 for gears of the same handedness

 for gears of opposite handedness

Where   is the helix angle for the gear. The crossed configuration is less

mechanically sound because there is only a point contact between the gears, whereas

in the parallel configuration there is a line contact.[9]

Quite commonly, helical gears are used with the helix angle of one having the

negative of the helix angle of the other; such a pair might also be referred to as having a

right-handed helix and a left-handed helix of equal angles. The two equal but opposite

angles add to zero: the angle between shafts is zero – that is, the shafts are parallel.

Where the sum or the difference (as described in the equations above) is not zero the

shafts are crossed. For shafts crossed at right angles, the helix angles are of the same

hand because they must add to 90 degrees.

DOUBLE HELICAL

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Double helical gears, or herringbone gears, overcome the problem of axial thrust

presented by "single" helical gears, by having two sets of teeth that are set in a V

shape. A double helical gear can be thought of as two mirrored helical gears joined

together. This arrangement cancels out the net axial thrust, since each half of the gear

thrusts in the opposite direction resulting in a net axial force of zero. This arrangement

can remove the need for thrust bearings. However, double helical gears are more

difficult to manufacture due to their more complicated shape.

For both possible rotational directions, there exist two possible arrangements for

the oppositely-oriented helical gears or gear faces. One arrangement is stable, and the

other is unstable. In a stable orientation, the helical gear faces are oriented so that each

axial force is directed toward the center of the gear. In an unstable orientation, both

axial forces are directed away from the center of the gear. In both arrangements, the

total (or net) axial force on each gear is zero when the gears are aligned correctly. If the

gears become misaligned in the axial direction, the unstable arrangement will generate

a net force that may lead to disassembly of the gear train, while the stable arrangement

generates a net corrective force. If the direction of rotation is reversed, the direction of

the axial thrusts is also reversed, so a stable configuration becomes unstable, and vice

versa.

Stable double helical gears can be directly interchanged with spur gears without

any need for different bearings.

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BEVEL GEAR

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 mitre

gears.

TYPES OF BEVEL GEARS11 | P a g e

Straight bevel gears have conical pitch surface and teeth are straight and tapering

towards apex.

Spiral bevel gears have curved teeth at an angle allowing tooth contact to be

gradual and smooth.

Zerol bevel gears are very similar to a bevel gear only exception is the teeth are

curved: the ends of each tooth are coplanar with the axis, but the middle of each

tooth is swept circumferentially around the gear. Zerol bevel gears can be thought of

as spiral bevel gears (which also have curved teeth) but with a spiral angle of zero

(so the ends of the teeth align with the axis).

Hypoid bevel gears are similar to spiral bevel but the pitch surfaces

are hyperbolic and not conical. Pinion can be offset above, or below,the gear

centre, thus allowing larger pinion diameter, and longer life and smoother mesh, with

additional ratios e.g., 6:1, 8:1, 10:1. In a limiting case of making the "bevel" surface

parallel with the axis of rotation, this configuration resembles a worm drive.

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GEOMETRY OF A BEVEL GEAR

A double-helical bevel gear made by Citroen in 1927 for the Miřejovice water

power plant

The cylindrical gear tooth profile corresponds to an involute, whereas the bevel

gear tooth profile is an octoid. All traditional bevel gear generators (such as 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 those "involute bevel gear sets" causes more noise.

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LIST OF DRAWING SYMBOLS

Np - Number of teeth on pinion

Ng - Number of teeth on given gear

Dg - Pitch diameter of given gear

Dp - Pitch diameter of given pinion

F - Face width (length of single tooth)

γ - Pinion pitch angle (radians)

Γ - Gear pitch angle (radians)

Ao - Cone distance (distance from pitch circle to intersection of shaft axes)

rb - Back-cone radius

P - Diametrical pitch (teeth per inch of pitch diameter (N/D))

p - Circular pitch (inches of circumference per tooth (Π/P))

Tooth shape for bevel gears is determined by scaling spur gear tooth shapes along the

face width. The further from the intersection of the gear and pinion axes, the bigger the

tooth cross sections are. If the tooth face were to extend all the way to the axes

intersection, the teeth would approach infinitesimal size there. The tooth cross-section

at the largest part of the tooth is identical to the tooth cross-section of a tooth from a

spur gear with Pitch Diameter of 2* rb, or twice the Back-Cone Radius, and with an

imaginary number of teeth (N’) equal to  times the Back-Cone Radius (rb) divided by

the Circular Pitch of the bevel gear (p). This method of obtaining the dimensions and

shape of the largest tooth profile is known at the “Tredgold” tooth-shape approximation.

Refer to the profiles shown near the Back-cone radius dimension in the drawing above.

Mean radius:

Hp=Tx n/63000 → T = Hp x 63000/n

T = Rm x Wt → Wt = Hp x 63000/ n x Rm

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TEETH/TOOTHLINE OF BEVEL GEARS

TEETH

There are two issues regarding tooth shape. One is the cross-sectional profile of

the individual tooth. The other is the line or curve on which the tooth is set on the face of

the gear: in other words the line or curve along which the cross-sectional profile is

projected to form the actual three-dimensional shape of the tooth. The primary effect of

both the cross-sectional profile and the tooth line or curve is on the smoothness of

operation of the gears. Some result in a smoother gear action than others.

TOOTH LINE

The teeth on bevel gears can be straight, spiral or "zerol".

STRAIGHT TOOTH LINES

In straight bevel gears the teeth are straight and parallel to the generators of

the cone. This is the simplest form of bevel gear. It resembles a spur gear, only conical

rather than cylindrical. The gears in the floodgate picture are straight bevel gears. In

straight bevel gear sets, when each tooth engages it impacts the corresponding tooth

and simply curving the gear teeth can solve the problem.

SPIRAL TOOTH LINES

Spiral bevel gears have their teeth formed along spiral lines. They are somewhat

analogous to cylindrical type helical gears in that the teeth are angled; however, with

spiral gears the teeth are also curved.

The advantage of the spiral tooth over the straight tooth is that they engage more

gradually. The contact between the teeth starts at one end of the gear and then spreads

across the whole tooth. This results in a less abrupt transfer of force when a new pair of

teeth come into play. With straight bevel gears, the abrupt tooth engagement causes

noise, especially at high speeds, and impact stress on the teeth which makes them

unable to take heavy loads at high speeds without breaking. For these reasons straight

bevel gears are generally limited to use at linear speeds less than 1000 feet/min; or, for

small gears, under 1000 r.p.m.[1]

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ZEROL TOOTH LINES

Zerol bevel gears are an intermediate type between straight and spiral bevel

gears. Their teeth are curved, but not angled. Zerol bevel gears are designed with the

intent of duplicating the characteristics of a straight bevel gear but they are produced

using a spiral bevel cutting process.

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HYPOID GEARS

Hypoid spiral bevel gears

A hypoid is a type of spiral bevel gear whose axis does not intersect with the

axis of the meshing gear. The shape of a hypoid gear is a revolved hyperboloid (that is,

the pitch surface of the hypoid gear is a hyperbolic surface), whereas the shape of a

spiral bevel gear is normally conical. The hypoid gear places the pinion off-axis to

the crown wheel (ring gear) which allows the pinion to be larger in diameter and have

more contact area. In hypoid gear design, the pinion and gear are practically always of

opposite hand, and the spiral angle of the pinion is usually larger than that of the gear.

The hypoid pinion is then larger in diameter than an equivalent bevel pinion.

A hypoid gear incorporates some sliding and can be considered halfway between a

straight-cut gear and a worm gear. Special gear oilsare required for hypoid gears

because the sliding action requires effective lubrication under extreme

pressure between the teeth.

Hypoid gearings are used in power transmission products that are more efficient than

conventional worm gearing.[citation needed] They are considerably stronger in that any load is

conveyed through multiple teeth simultaneously. By contrast, bevel gears are loaded

through one tooth at a time. The multiple contacts of hypoid gearing, with proper

lubrication, can be nearly silent, as well.

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CROWN GEAR

A crown gear (or a contrate gear) is a gear which has teeth that project at right angles to the face of the wheel. In particular, a crown gear is a type of bevel gear where the pitch cone angle is 90 degrees.[1][2] A pitch cone of any other angle is simply called a bevel gear.[3] Crown gears normally mesh with other bevel gears, or sometimes spur gears.

18 | P a g e

HELICAL GEARS

The teeth on helical gears are cut at an angle to the face of the gear. When two teeth

on a helical gear system engage, the contact starts at one end of the tooth and

gradually spreads as the gears rotate, until the two teeth are in full engagement.

This gradual engagement makes helical gears operate much more smoothly and quietly

than spur gears. For this reason, helical gears are used in almost all car transmissions.

Because of the angle of the teeth on helical gears, they create a thrust load on the gear

when they mesh. Devices that use helical gears have bearings that can support this

thrust load.

One interesting thing about helical gears is that if the angles of the gear teeth are

correct, they can be mounted on perpendicular shafts, adjusting the rotation angle by 90

degrees.

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HELICAL GEAR PARAMETERS

A helical gear train with parallel axes is very similar to a spur

gear with the same tooth profile and proportions.  The primary

difference is that the teeth are machined at an angle to the

gear axis.

Helix angle  

The helix angle of helical gears β is generally selected from the range

6,8,10,12,15,20 degrees. The larger the angle the smoother the motion and the higher

speed possible however the thrust loadings on the supporting bearings also increases.  

In case of a double or herringbone gear β values 25,30,35,40 degrees can also be

used.  These large angles can be used because the side thrusts on the two sets of teeth

cancel each other allowing larger angles with no penalty.

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Pitch /module

For helical gears the circular pitch is measured in two ways

The traverse circular pitch (p) is the same as for spur gears and is measured along the

pitch circle

The normal circular pitch p n is measured normal to the helix of the gear.

The diametric pitch is the same as for spur gears     ... P = z g /dg = z p /d p ....d= pitch

circle dia (inches). 

The module is the same as for spur gears     ... m = dg/z g = d p/z p.... d = pitch circle dia

(mm).

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HELICAL GEAR GEOMETRICAL PROPORTIONS

p = Circular pitch = d g.  / z g = d p.  / z p

p n = Normal circular pitch = p .cosβ

P n =Normal diametrical pitch = P /cosβ

p x = Axial pitch = p c /tanβ

m n =Normal module = m / cosβ

α n = Normal pressure angle = tan -1 ( tanα.cos β )

β =Helix angle

d g = Pitch diameter gear = z g. m

d p = Pitch diameter pinion = z p. m

a =Center distance = ( z p + z g )* m n /2 cos β

a a = Addendum = m

a f =Dedendum = 1.25*m

b = Face width of narrowest gear

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HERRINGBONE / DOUBLE CROSSED HELICAL

GEARS

A herringbone gear, a specific type of double helical gear,[1] is a special type of gear which is a side to side (not face to face)

combination of two helical gears of opposite hands.[2] From the top

the helical grooves of this gear looks like letter V. Unlike helical

gearsthey do not produce an additional axial load.

Like helical gears, they have the advantage of transferring

power smoothly because more than two teeth will be in mesh at any

moment in time. Their advantage over the helical gears is that the

side-thrust of one half is balanced by that of the other half. This

means that herringbone gears can be used in torque gearboxes

without requiring a substantial thrust bearing. Because of

this herringbone gears were an important step in the introduction of

the steam turbine to marine propulsion.

Precision herringbone gears are more difficult to

manufacture than equivalent spur or helical gears and consequently

are more expensive. They are used in heavy machinery.

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CROSSED HELICAL GEARS

When two helical gears are used to transmit power between

non parallel, non-intersecting shafts, they are generally called

crossed helical gears.  These are simply normal helical gears with

non-parallel shafts.  For crossed helical gears to operate successfully

they must have the same pressure angle and the same normal pitch.

They need not have the same helix angle and they do not need to be

opposite hand.   The contact is not a good line contact as for parallel

helical gears and is often little more than a point contact.  Running in

crossed helical gears tend to marginally improve the area of contact.

The relationship between the shaft angles E and the helix angles

β 1 & β2 is as follows

E = (Same Helix Angle) β 1 + β 2 ......(Opposite Helix

Angle) β 1 - β 2

For gears with a 90o crossed axis it is obvious that the

gears must be the same hand.The centres distance (a) between

crossed helical gears is calculated as follows

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a = m * [(z 1 / cos β 1) + ( z 1 / cos β 1 )] / 2

The sliding velocity Vsof crossed helical gears is given by

Vs = (V1 / cos β 1 ) = (V 2 / cos β 2 )

STRENGTH AND DURABILITY CALCULATIONS

FOR HELICAL GEAR TEETH

Designing helical gears is normally done in accordance with

standards the two most popular series are listed under

standards above: The notes below relate to approximate

methods for estimating gear strengths. The methods are really

only useful for first approximations and/or selection of stock

gears (ref links below). � Detailed design of spur and helical

gears should best be completed using :

a) Standards. 

b) Books are available providing the necessary guidance. 

c) Software is also available making the process very easy.    A

very reasonably priced and easy to use package is included in

the links below (Mitcalc.com)

The determination of the capacity of gears to transfer the required

torque for the desired operating life is completed by

determining the strength of the gear teeth in bending and also

the durability i.e of the teeth ( resistance to

wearing/bearing/scuffing loads ) .. The equations below are

25 | P a g e

based on methods used by Buckingham..

Bending 

The Lewis formula for spur gears can be applied to helical gears

with minor adjustments to provide an initial conservative

estimate of gear strength in bending.    This equation should

only be used for first estimates. 

σ = Fb / ( ba. m. Y )

Fb = Normal force on tooth = Tangential Force

Ft / cos β

σ = Tooth Bending stress (MPa)

ba = Face width (mm)

Y = Lewis Form Factor

m = Module (mm)

When a gear wheel is rotating the gear teeth come into contact

with some degree of impact.  To allow for this a velocity factor

is introduced into the equation.   This is given by the Barth

equation for milled profile gears.

K v = (6,1 + V ) / 6,1

26 | P a g e

V = the pitch line velocity = PCD.w/2 

The Lewis formula is thus modified as follows

σ = K v.Fb / ba. m. Y

The Lewis form factor Y must be determined for the virtual number

of teeth z' = z /cos3β The bending stress resulting should be

less than the allowable bending stress Sb for the gear material

under consideration.   Some sample values are provide on this

page ef Gear Strength Values

Surface Strength 

The allowable gear force from surface durability considerations is

determined approximately using the simple equation as follows

Fw = K v d p b a Q K / cos2β

Q = 2. dg /( dp + dp ) = 2.zg /( zp +zp )

Fw = The allowable gear load. (MPa)

K = Gear Wear Load Factor (MPa) obtained by look up

ref Gear Strength Values

27 | P a g e

Lewis Form factor for Teeth profile α = 20o , addendum = m,

dedendum = 1.25m

Numb

er of

teeth

Y

Numb

er of

teeth

Y

Numb

er of

teeth

Y

Numb

er of

teeth

Y

Numb

er of

teeth

Y

120.24

517

0.30

322

0.33

134

0.37

175

0.43

5

130.26

118

0.30

924

0.33

738

0.38

4100

0.44

7

140.27

719

0.31

426

0.34

645

0.40

1150

0.46

0

150.29

020

0.32

228

0.35

350

0.40

9300

0.47

2

28 | P a g e

160.29

621

0.32

830

0.35

960

0.42

2Rack

0.48

5

WORM GEAR

29 | P a g e

Worm gears are used when large gear reductions are needed. It is common for

worm gears to have reductions of 20:1, and even up to 300:1 or greater.

Many worm gears have an interesting property that no other gear set has: the

worm can easily turn the gear, but the gear cannot turn the worm. This is because the

angle on the worm is so shallow that when the gear tries to spin it, the friction between

the gear and the worm holds the worm in place.

This feature is useful for machines such as conveyor systems, in which the

locking feature can act as a brake for the conveyor when the motor is not turning. One

other very interesting usage of worm gears is in the Torsen differential, which is used on

some high-performance cars and trucks.

A worm gear is used when a large speed reduction ratio is required between

crossed axis shafts which do not intersect.   A basic helical gear can be used but the

power which can be transmitted is low.  A worm drive consists of a large diameter worm

30 | P a g e

wheel with a worm screw meshing with teeth on the periphery of the worm wheel.   The

worm is similar to a screw and the worm wheel is similar to a section of a nut.   As the

worm is rotated the worm wheel is caused to rotate due to the screw like action of the

worm.  The size of the worm gearset is generally based on the centre distance between

the worm and the worm wheel.

If the worm gears are machined basically as crossed helical gears the result is a highly

stress point contact gear.  However normally the wormwheel is cut with a concave as

opposed to a straight width.   This is called a single envelope worm gearset.   If the

worm is machined with a concave profile to effectively wrap around the wormwheel the

gearset is called a double enveloping worm gearset and has the highest power capacity

for the size.  Single enveloping gearsets require accurate alignment of the worm-wheel

to ensure full line tooth contact. Double enveloping gearsets require accurate alignment

of both the worm and the wormwheel to obtain maximum face contact.

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Diagram showing the different worm gear options available.

The double enveloping (double throat/double globoid ) option is the most difficult

to manufacture and set up.     However this option has the highest load capacity, near

zero backlash capability, highest accuracy and extended life capability.

32 | P a g e

A more detailed view showing a cylinderical worm and an enveloping gear.    The

worm is shown with the worm above the wormwheel.  The gearset can also be arranged

with the worm below the wormwheel.   Other alignments are used less frequently.

33 | P a g e

NOMENCLATURE OF WORM GEAR

As can be seen in the above view a section through the axis of the worm and the

centre of the gear shows that , at this plane, the meshing teeth and thread section is

similar to a spur gear and has the same features

αn = Normal pressure angle = 20o as standard 

γ = Worm lead angle = (180 /π ) tan-1 (z 1 / q)(deg)   ..Note: for α n= 20o  γ should be less

than 25o

b a = Effective face width of worm wheel. About 2.m √ (q +1) (mm)

b l = Length of worm wheel. About 14.m. (mm)

c = clearance   c min = 0,2.m cos γ ,  c max = 0,25.m cos γ (mm) 

d 1 = Ref dia of worm (Pitch dia of worm (m)) = q.m (mm)

d a.1 = Tip diameter of worm = d 1 + 2.h a.1 (mm)

d 2 = Ref dia of worm wheel (Pitch dia of wormwheel) =( p x.z/π ) = 2.a - d 1 (mm)

d a.2 = Tip dia worm wheel (mm)

h a.1 = Worm Thread addendum = m (mm)

h f.1 = Worm Thread dedendum , min = m.(2,2 cos γ - 1 ) , max = m.(2,25 cos γ - 1 )

(mm)

m = Axial module = p x /π (mm)

m n = Normal module = m cos γ(mm)

M 1 = Worm torque (Nm)

M 2 = Worm wheel torque (Nm)

n 1 = Rotational speed of worm (revs /min)

n 2 = Rotational speed of wormwheel (revs /min)

p x = Axial pitch of of worm threads and circular pitch of wheel teeth ..the pitch between

adjacent threads = π. m. (mm)

p n = Normal pitch of of worm threads and gear teeth (m)

q = Worm diameter factor = d 1 / m - (Allows module to be applied to worm ) selected

from (6   6,5   7   7,5   8   8,5   9  10   11   12   13   14   17   20 )

34 | P a g e

p z = Lead of worm = p x. z 1 (mm).. Distance the thread advances in one rev'n of the

worm.   For a 2-start worm the lead = 2 . p x

R g = Reduction Ratio

μ = coefficient of friction

η= Efficiency

Vs = Worm-gear sliding velocity ( m/s)

z 1 = Number of threads (starts) on worm

z 2 = Number of teeth on wormwheel

35 | P a g e

WORM GEAR DESIGN PARAMETERS

Worm gears provide a normal single reduction range of 5:1 to 75-1.  The pitch

line velocity is ideally up to 30 m/s.  The efficiency of a worm gear ranges from 98% for

the lowest ratios to 20% for the highest ratios.  As the frictional heat generation is

generally high the worm box must be designed disperse heat to the surroundings and

lubrication is an essential requirement.  Worm gears are quiet in operation.  Worm

gears at the higher ratios are inherently self locking - the worm can drive the gear but

the gear cannot drive the worm.   A worm gear can provide a 50:1 speed reduction but

not a 1:50 speed increase....(In practice a worm should not be used a braking device for

safety linked systems e.g hoists.  . Some material and operating conditions can result in

a wormgear backsliding )

The worm gear action is a sliding action which results in significant frictional losses.  

The ideal combination of gear materials is for a case hardened alloy steel worm (ground

finished) with a phosphor bronze gear.  Other combinations are used for gears with

comparatively light loads.

SPECIFICATIONS OF WORM GEAR

BS721 Pt2 1983 Specification for worm gearing � Metric units. 

This standard is current (2004) and provides information on tooth form, dimensions of

gearing, tolerances for four classes of gears according to function and accuracy,

calculation of load capacity and information to be given on drawings.

36 | P a g e

WORM GEAR DESIGNATION

Very simply a pair of worm gears can be defined by designation of the number of

threads in the worm ,the number of teeth on the wormwheel, the diameter factor and the

axial module i.e z1,z2, q, m .

This information together with the centre distance ( a ) is enough to enable calculation

of and any dimension of a worm gear using the formulea available.

WORM TEETH PROFILE

The sketch below shows the normal (not axial) worm tooth profile as indicated in

BS 721-2 for unit axial module (m = 1mm) other module teeth are in proportion e.g.

2mm module teeth are 2 times larger

Typical axial modules values (m) used for worm gears are

0,5    0,6     0,8    1,0     1,25    1,6    2,0    2,5    3,15     4,0     5,0     6,3    8,0    10,0    

12,5    16,0    20,0    25,0    32,0    40,0    50,0

37 | P a g e

MATERIALS USED FOR GEARS

Material Notes applications

Worm

Acetal / Nylon Low Cost, low dutyToys, domestic appliances,

instruments

Cast IronExcellent machinability,

medium friction.

Used infrequently in

modern machinery

Carbon SteelLow cost, reasonable

strength

Power gears with medium

rating.

Hardened SteelHigh strength, good

durability

Power gears with high

rating for extended life

Wormwheel

Acetal /Nylon Low Cost, low dutyToys, domestic appliances,

instruments

Phos Bronze

Reasonable strength, low

friction and good

compatibility with steel

Normal material for worm

gears with reasonable

efficiency

Cast IronExcellent machinability,

medium friction.

Used infrequently in

modern machinery

BACKLASH / QUALITY GRADES

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A worm gear set normally includes some backlash during normal manufacture to

allow for expansion of the gear wheel when operating at elevated temperaturs.     The

backlash is controlled by adusting the gear wheel tooth thickness.

BS 721 includes a table of backlash limits related to the accuracy grade.    The standard

lists 5 accuracy grades.

AGMA and DIN provide a similar grading system

Grade 1 relates to critical applications where minimum backlash

is required i.e instruments /metering

Grade 2 relates to precision drives such as machine tools

Grade 3,4,5 relates to industrial drives with working

temperatures of about 120o C

INITIAL SIZING OF WORM GEAR (MECHANICAL)

39 | P a g e

1) Initial information generally Torque required (Nm), Input speed(rpm), Output speed

(rpm).

2) Select Materials for worm and wormwheel.

3) Calculate Ratio (R g)

4) Estimate a = Center distance (mm)

5) Set z 1 = Nearest number to (7 + 2,4 SQRT (a) ) /R g 

6) Set z 2 = Next number < R g . z 1

7) Using the value of estimated centre distance (a) and No of gear teeth ( z 2 )  obtain a

value for q from the table below. (q -value selection)

8) d 1 = q.m (select) ..

9) d 2 = 2.a - d 1

10) Select a wormwheel face width b a (minimum =2*m*SQRT(q+1))

11) Calculate the permissible output torques for strength (M b_1 and wear M c_1 )

12) Apply the relevent duty factors to the allowable torque and the actual torque

13) Compare the actual values to the permissible values and repeat process if

necessary

14) Determine the friction coefficient and calculate the efficiency.

15) Calculate the Power out and the power in and the input torque

16) Complete design of gearbox including design of shafts, lubrication, and casing

ensuring sufficient heat transfer area to remove waste heat.

INITIAL SIZING OF WORM GEAR (THERMAL)

40 | P a g e

Worm gears are often limited not by the strength of the teeth but by the heat

generated by the low efficiency. It is necessary therefore to determine the heat

generated by the gears = (Input power - Output power). The worm gearbox must have

lubricant to remove the heat from the teeth in contact and sufficient area on the external

surfaces to distibute the generated heat to the local environment. This requires

completing an approximate heat transfer calculation. If the heat lost to the environment

is insufficient then the gears should be adjusted (more starts, larger gears) or the box

geometry should be adjusted, or the worm shaft could include a fan to induced forced

air flow heat loss.

FORMULAE

41 | P a g e

The reduction ratio of a worm gear ( R g )

R g = z 2 / z 1

eg a 30 tooth wheel meshing with a 2 start worm has a reduction of 15 

Tangential force on worm ( F wt )= axial force on wormwheel

F wt = F ga = 2.M 1 / d 1

Axial force on worm ( F wa ) = Tangential force on gear

Output torque ( M 2 ) = Tangential force on wormwheel * Wormwheel reference

diameter /2

42 | P a g e

M 2 = F gt* d 2 / 2

Relationship between the Worm Tangential Force F wt and the Gear Tangential

force F gt

Relationship between the output torque M 2and the input torque M 1

M 2 = ( M 1. d 2 / d 1 ).[ (cos α n - μ tan γ ) / (cos α n . tan (γ + μ) ) ]

Separating Force on worm-gearwheel ( F s )

Sliding velocity ( V s )...(m/s)

V s (m/s ) = 0,00005236. d 1. n 1 sec γ 

= 0,00005235.m.n (z 12 + q 2 ) 1/2

Peripheral velocity of wormwheel ( V p) (m/s) 

V p = 0,00005236,d 2. n 2

43 | P a g e

FRICTION COEFFICIENT

Note: The values of the coeffient of friction as provided in the table below are

based on the use of phosphor bronze wormwheels and case hardended , ground and

polished steel worms , lubricated by a mineral oil having a viscosity of between 60cSt,

and 130cSt at 60 deg.C .

Cast Iron and Phosphor Bronze .. Table x 1,15 

Cast Iron and Cast Iron.. Table x 1,33 

Quenched Steel and Aluminum Alloy..Table x 1,33 

Steel and Steel..Table x 2

Friction coefficients - For Case Hardened Steel Worm / Phos Bros Wheel

Sliding

Speed

Friction

Coefficient

Sliding

Speed

Friction

Coefficient

m/s μ m/s μ

0 0,145 1,5 0,038

0,001 0,12 2 0,033

0,01 0,11 5 0,023

0,05 0,09 8 0,02

0,1 0,08 10 0,018

0,2 0,07 15 0,017

0,5 0,055 20 0,016

1 0,044 30 0,016

44 | P a g e

EFFICIENCY OF WORM GEAR  

The efficiency of the worm gear is determined by dividing the output Torque M2

with friction = μ by the output torque with zero losses i.e μ = 0

First cancelling [( M 1. d 2 / d 1 ) / M 1. d 2 / d 1 ) ] = 1 

Denominator = [(cos α n / (cos α n . tan γ ] = cot γ

η = [(cos α n - μ tan γ ) / (cos α n . tan γ + μ ) ] / cot γ

= [(cos α n - μ .tan γ ) / (cos α n + μ .cot γ )]

Graph showing worm gear efficiency related to gear lead angle ( γ )

45 | P a g e

SELF LOCKING

Referring to the above graph , When the gear wheel is driving the curve points

intersecting the zero efficiency line identify when the worm drive is self locking i.e the

gear wheel cannot drive to worm.    It is the moment when gearing cannot be moved

using even the highest possible torque acting on the worm gear.    The self-locking limit

occurs when the worm lead angle ( γ ) equals atan (μ). (2o to 8o )

It is often considered that the static coefficient of friction is most relevant as the gear

cannot be started.    However in practice it is safer to use the, lower, dynamic coefficient

of friction as this comes into play if the gear set is subject to vibration.

Worm Design /Gear Wear / Strength Equations to BS721

Note: For designing worm gears to AGMA codes AGMA method of Designing

Worm Gears 

The information below relates to BS721 Pt2 1983 Specification for worm gearing �

Metric units.  BS721 provides average design values reflecting the experience of

specialist gear manufacturers.   The methods have been refined by addition of various

application and duty factors as used.  Generally wear is the critical factor.

46 | P a g e

PERMISSIBLE LOAD FOR STRENGTH  

The permissible torque (M in Nm) on the gear teeth is obtained by use of the

equation

M b = 0,0018 X b.2σ bm.2. m. l f.2. d 2.

( example 87,1 Nm = 0,0018 x 0,48 x 63 x 20 x 80 ) 

X b.2 = speed factor for bending (Worm wheel ).. See Below

σ bm.2 = Bending stress factor for Worm wheel.. See Table below

l f.2 = length of root of Worm Wheel tooth

d 2 = Reference diameter of worm wheel

m = axial module

γ = Lead angle

PERMISSIBLE TORQUE FOR WEAR  

The permissible torque (M in Nm) on the gear teeth is obtained by use of the

equation

M c = 0,00191 X c.2σ cm.2.Z. d 21,8. m

( example 33,42 Nm = 0,00191 x 0,3234 x 6,7 x 1,5157 x 801,8 x 2 ) 

X c.2 = Speed factor for wear ( Worm wheel )

σ cm.2 = Surface stress factor for Worm wheel

Z = Zone factor.

47 | P a g e

LENGTH OF ROOT OF WORM WHEEL TOOTH  

Radius of the root = R r= d 1 /2 + h ha,1 (= m) + c(= 0,25.m.cos γ )

R r= d 1 /2 + m(1 +0,25 cosγ) 

l f.2 = 2.R r.sin-1 (2.R r / b a)

Note: angle from sin-1(function) is in radians...

Speed Factor for Bending 

This is a metric conversion from an imperial formula..

X b.2 = speed factor for bending = 0,521(V) -0,2

V= Pitch circle velocity =0,00005236*d 2.n 2 (m/s)

The table below is derived from a graph in BS 721. I cannot see how

this works as a small worm has a smaller diameter compared to a

large worm and a lower speed which is not reflected in using the

RPM. 

Table of speed factors for bending 

RPM (n2) X b.2 RPM (n2) X b.2

1 0,62 600 0,3

10 0,56 1000 0,27

20 0,52 2000 0,23

60 0,44 4000 0,18

100 0,42 6000 0,16

200 0,37 8000 0,14

400 0,33 10000 0,13

48 | P a g e

Additional factors

The formula for the acceptable torque for wear should be modified to allow

additional factors which affect the Allowable torque M c

M c2 = M c. Z L. Z M.Z R / K C

The torque on the wormwheel as calculated using the duty requirements (M e)

must be less than the acceptable torque M c2 for a duty of 27000 hours with uniform

loading.   For loading other than this then M e should be modified as follows

M e2 = M e. K S* K H

Thus 

uniform load < 27000 hours (10 years) M e ≤ M c2 

Other conditions M e2 ≤ M c2 

Factors used in equations

Lubrication (Z L)..

Z L = 1 if correct oil with anti-scoring additive else a lower value should be selected

Lubricant (Z M).. 

Z L = 1 for Oil bath lubrication at V s < 10 m /s 

Z L = 0,815 Oil bath lubrication at 10 m/s < V s < 14 m /s 

Z L = 1 Forced circulation lubrication 

Surface roughness (Z R ) ..

Z R = 1 if Worm Surface Texture < 3μ m and Wormwheel < 12 μ m 

else use less than 1

49 | P a g e

Tooth contact factor (K C

This relates to the quality and rigidity of gears . Use 1 for first estimate

K C = 1 For grade A gears with > 40% height and > 50% width contact 

= 1,3 - 1,4 For grade A gears with > 30% height and > 35% width contact 

= 1,5-1,7 For grade A gears with > 20% height and > 20% width contact 

Starting factor (K S) ..

K S =1 for < 2 Starts per hour

=1,07 for 2- 5 Starts per hour

=1,13 for 5-10 Starts per hour

=1,18 more than 10 Starts per hour

Time / Duty factor (K H) .. 

K H for 27000 hours life (10 years) with uniform driver and driven loads

SPEED FACTORS  

50 | P a g e

X c.2 = K V .K R 

Note: This table is not based on the graph in BS 721-2 (figure 7) it is based on another

more easy to follow graph.   At low values of sliding velocity and RPM it agrees closely

with BS 721.   At higher speed velocities it gives a lower value (e.g at 20m/s -600 RPM

the value from this table for X c.2 is about 80% of the value in BS 721-2

Table of Worm Gear Speed Factors 

Note -sliding speed = Vs and Rotating speed = n2 (Wormwheel)

Sliding speed K V Rotating Speed K R

m/s rpm

0 1 0,5 0,98

0,1 0,75 1 0,96

0,2 0,68 2 0,92

0,5 0,6 10 0,8

1 0,55 20 0,73

2 0,5 50 0,63

5 0,42 100 0,55

10 0,34 200 0,46

20 0,24 500 0,35

30 0,16 600 0,33

WORM Q VALUE SELECTION

The table below allows selection of q value which provides a reasonably efficient

worm design.   The recommended centre distance value "a" (mm) is listed for each q

51 | P a g e

value against a range of z 2 (teeth number values).   The table has been produced by

reference to the relevant plot in BS 721 

Example

If the number of teeth on the gear is selected as 45 and the centre distance is 300 mm

then a q value for the worm would be about 7.5

Important note: This table provides reasonable values for all worm speeds. However at

worm speeds below 300 rpm a separate plot is provided in BS721 which produces more

accurate q values.    At these lower speeds the resulting q values are approximately 1.5

higher than the values from this table. The above example at less than 300rpm should

be increased to about 9

Table of Center distances "a" relating to q values and Number of teeth on Worm

gear z 2

Number of Teeth On Worm Gear (z 2)

q 20 25 30 35 40 45 50 55 60 65 70 75 80

6 150 250 380 520 700

6.5 100 150 250 350 480 660

7 70 110 170 250 350 470 620 700

7.5 50 80 120 180 240 330 420 550 670

8 25 50 80 120 180 230 300 380 470 570 700

8.5 28 90 130 130 180 220 280 350 420 500 600 700

9 40 70 100 130 170 220 280 330 400 450 520

9.5 25 50 70 100 120 150 200 230 300 350 400

10 26 55 80 100 130 160 200 230 270 320

11 25 28 55 75 100 130 150 180 220 250

12 28 45 52 80 100 130 150 100

13 27 45 52 75 90 105

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AGMA METHOD OF DESIGNING WORM GEARS

The AGMA method is provided here because it is relatively easy to use and

convenient- AGMA is all imperial and so I have used conversion values so all

calculations can be completed in metric units.. 

Good proportions indicate that for a centre to centre distance = C the mean worm dia

d 1 is within the range 

Imperial (inches)

( C 0,875 / 3 )   ≤  d 1    ≤   ( C 0,875 / 1,6 )

Metric ( mm)

( C 0,875 / 2 )   ≤  d 1   ≤   ( C 0,875 / 1,07 )

The acceptable tangential load (W t) all

(W t) all = C s. d 20,8 .b a .C m .C v . (0,0132) (N)

The formula will result in a life of over 25000 hours with a case hardened alloy

steel worm and a phosphor bronze wheel

C s = Materials factor

b a = Effective face width of gearwheel = actual face width. but not to exceed 0,67 . d 1

C m = Ratio factor 

C v = Velocity factor

Modified Lewis equation for stress induced in worm gear teeth .

σ a = W t / ( p n. b a. y )(N)

53 | P a g e

W t = Worm gear tangential Force (N)

y = 0,125 for a normal pressure angle α n = 20o

The friction force = W f

W f = f.W t / (. cos φ n ) (N)

γ = worm lead angle at mean diameter

α n = normal pressure angle

The sliding velocity = V s

V s = π .n 1. d 1 / (60,000 )

d 1 = mean dia of worm (mm)

n 1 = rotational speed of worm (revs/min)

The torque generated γ at the worm gear = M b (Nm)

T G = W t .d 1 / 2000

The required friction heat loss from the worm gearbox

H loss = P in ( 1 - η )

η = gear efficiency as above.

C s values

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C s = 270 + 0,0063(C )3... for C ≤ 76mm ....Else 

C s (Sand cast gears ) = 1000 for d 1 ≤ 64 mm ...else... 1860 - 477 log (d 1 )

C s (Chilled cast gears ) = 1000 for d 1 ≤ 200 mm ...else ... 2052 -456 log (d 1 )

C s (Centrifugally cast gears ) = 1000 for d 1 ≤ 635 mm ...else ... 1503 - 180 log (d 1 )

C m values

NG = Number of teeth on worm gear.

NW = Number of starts on worm gear.

mG = gear ration = NG /NW

C v values

C v (V s > 3,56 m/s ) = 0,659 exp (-0,2167 V s ) 

C v (3,56 m/s ≤ V s < 15,24 m/s ) = 0,652 (V s) -0,571 ) 

C v (V s > 15,24 m/s ) = 1,098.( V s ) -0,774 )

f values

55 | P a g e

f (V s = 0) = 0,15 

f (0 < V s ≤ 0,06 m/s ) = 0,124 exp (-2,234 ( V s ) 0,645 

f (V s > 0,06 m/s ) = 0,103 exp (-1,1855 ( V s ) ) 0,450 ) +0,012

NON-CIRCULAR GEAR

56 | P a g e

Non-circular gear example

Another non-circular gear

A non-circular gear (NCG) is a special gear design with special characteristics

and purpose. While a regular gear is optimized to transmittorque to another engaged

57 | P a g e

member with minimum noise and wear and with maximum efficiency, a non-circular

gear's main objective might be ratio variations, axle displacement oscillations and more.

Common applications include textile machines,[1] potentiometers, CVTs (continuously

variable transmissions),[2] window shade panel drives, mechanical presses and high

torque hydraulic engines.[1]

A regular gear pair can be represented as two circles rolling together without slip.

In the case of non-circular gears, those circles are replaced with anything different from

a circle. For this reason NCGs in most cases are not round, but round NCGs looking

like regular gears are also possible (small ratio variations result from meshing area

modifications).

Generally NCG should meet all the requirements of regular gearing, but in some

cases, for example variable axle distance, could prove impossible to support and such

gears require very tight manufacturing tolerances and assembling problems arise.

Because of complicatedgeometry, NCGs are most likely spur

gears and molding or electrical discharge machining technology is used instead of

generation.

RACK AND PINION GEAR

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A rack and pinion is a type of linear actuator that comprises a pair of gears which

convert rotational motion into linear motion. A circular gear called "the pinion" engages

teeth on a linear "gear" bar called "the rack"; rotational motion applied to the pinion

causes the rack to move, thereby translating the rotational motion of the pinion into the

linear motion of the rack.

For example, in a rack railway, the rotation of a pinion mounted on a locomotive or

a railcar engages a rack between the rails and forces a train up a steep slope.

For every pair of conjugate involute profile, there is a basic rack. This basic rack is the

profile of the conjugate gear of infinite pitch radius.[1] (I.e. a toothed straight edge.)

A generating rack is a rack outline used to indicate tooth details and dimensions for the

design of a generating tool, such as a hob or a gear shaper cutter.[1]

Rack and pinion combinations are often used as part of a simple linear actuator, where

the rotation of a shaft powered by hand or by a motor is converted to linear motion.

59 | P a g e

The rack carries the full load of the actuator directly and so the driving pinion is usually

small, so that the gear ratio reduces the torque required. This force, thus torque, may

still be substantial and so it is common for there to be a reduction gear immediately

before this by either a gear orworm gear reduction. Rack gears have a higher ratio, thus

require a greater driving torque, than screw actuators

EPICYCLIC GEAR

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An epicyclic gear train consists of two gears mounted so that the center of one gear

revolves around the center of the other. A carrier connects the centers of the two gears

and rotates to carry one gear, called the planet gear, around the other, called the sun

gear. The planet and sun gears mesh so that their pitch circles roll without slip. A point

on the pitch circle of the planet gear traces an epicycloid curve. In this simplified case,

the sun gear is fixed and the planetary gear(s) roll around the sun gear.

An epicyclic gear train can be assembled so the planet gear rolls on the inside of the

pitch circle of a fixed, outer gear ring, which is called an annular gear. In this case, the

curve traced by a point on the pitch circle of the planet is a hypocycloid.

The combination of epicycle gear trains with a planet engaging both a sun gear and an

annular gear is called a planetary gear train.[1][2] In this case, the annular gear is usually

fixed and the sun gear is driven.

Epicyclic gears get their name from their earliest application, which was the modeling of

the movements of the planets in the heavens. Believing the planets, as everything in the

heavens, to be perfect, they could only travel in perfect circles, but their motions as

viewed from Earth could not be reconciled with circular motion. At around 500 BC, the

Greeks invented the idea of epicycles, of circles traveling on the circular orbits. With this

theory Claudius Ptolemy in the Almagest in 148 AD was able to predict planetary orbital

paths. The Antikythera Mechanism, circa 80 BC, had gearing which was able to

approximate the moon's elliptical path through the heavens, and even to correct for the

nine-year precession of that path.[3] (Of course, the Greeks would have seen it as not

elliptical, but rather epicyclic, motion.)

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Epicyclic gearing or planetary gearing is a gear system consisting of one or more

outer gears, or planet gears, revolving about a central, or sun gear. Typically, the planet

gears are mounted on a movable arm or carrier which itself may rotate relative to the

sun gear. Epicyclic gearing systems also incorporate the use of an outer ring gear

orannulus, which meshes with the planet gears. Planetary gears (or epicyclic gears) are

typically classified as simple and compound planetary gears. Simple planetary gears

have one sun, one ring, one carrier, and one planet set. Compound planetary gears

involve one or more of the following three types of structures: meshed-planet (there are

at least two more planets in mesh with each other in each planet train), stepped-planet

(there exists a shaft connection between two planets in each planet train), and multi-

stage structures (the system contains two or more planet sets). Compared to simple

planetary gears, compound planetary gears have the advantages of larger reduction

ratio, higher torque-to-weight ratio, and more flexible configurations.

The axes of all gears are usually parallel, but for special cases like pencil

sharpeners and differentials, they can be placed at an angle, introducing elements

of bevel gear (see below). Further, the sun, planet carrier and annulus axes are

usually coaxial.

Epicyclic gearing is also available which consists of a sun, a carrier, and two planets

which mesh with each other. One planet meshes with the sun gear, while the second

planet meshes with the ring gear. For this case, when the carrier is fixed, the ring gear

62 | P a g e

rotates in the same direction as the sun gear, thus providing a reversal in direction

compared to standard epicyclic gearing.

History

In the 2nd-century AD treatise Almagest, Ptolemy used rotating deferent and

epicycles that form epicyclic gear trains to predict the motions of the planets. Accurate

predictions of the movement of the Sun, Moon and the five planets, Mercury, Venus,

Mars, Jupiter and Saturn, across the sky assumed that each followed a trajectory traced

by a point on the planet gear of an epicyclic gear train. This curve is called

an epitrochoid.

Epicyclic gearing was used in the Antikythera Mechanism, circa 80 BCE, to adjust the

displayed position of the moon for its ellipticity, and even for the precession of the

ellipticity. Two facing gears were rotated around slightly different centers, and one drove

the other not with meshed teeth but with a pin inserted into a slot on the second. As the

slot drove the second gear, the radius of driving would change, thus invoking a

speeding up and slowing down of the driven gear in each revolution.

Richard of Wallingford, an English abbot of St Albans monastery is credited for

reinventing epicyclic gearing for an astronomical clock in the 14th century.[4]

In 1588, Italian military engineer Agostino Ramelli invented the bookwheel, a vertically-

revolving bookstand containing epicyclic gearing with two levels of planetary gears to

maintain proper orientation of the books.[4][5]

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Gear ratio of Standard Epicyclic Gearing

In this example, the carrier (green) is held stationary while the sun gear (yellow) is used

as input. The planet gears (blue) turn in a ratio determined by the number of teeth in

each gear. Here, the ratio is −24/16, or −3/2; each planet gear turns at 3/2 the rate of

the sun gear, in the opposite direction.

The gear ratio of an epicyclic gearing system is somewhat non-intuitive, particularly

because there are several ways in which an input rotation can be converted into an

output rotation. The three basic components of the epicyclic gear are:

Sun: The central gear

Planet carrier: Holds one or more peripheral planet gears, all of the same size,

meshed with the sun gear

Annulus: An outer ring with inward-facing teeth that mesh with the planet gear or

gears

The overall gear ratio of a simple planetary gearset can be reliably calculated using the

following two equations,[6] representing the sun-planet and planet-annulus interactions

respectively:

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From which we can deduce that:

OR

Considering 

Where:

is the angular velocity of the Annulus, Sun Gear, Planet

Gears and Planet Carrier respectively.

is the Number of teeth of the Annulus, the Sun Gear and each Planet

Gear respectively.

Alternatively, if the number of teeth on each gear meets the

relationship  , this equation can be re-written as the following:

, where 

These relationships can be used to analyze any epicyclic system, including those, such

as hybrid vehicle transmissions, where two of the components are used as inputs with

the third providing output relative to the two inputs.[7]

In many epicyclic gearing systems, one of these three basic components is held

stationary; one of the two remaining components is an input, providing power to the

system, while the last component is an output, receiving power from the system. The

ratio of input rotation to output rotation is dependent upon the number of teeth in each

gear, and upon which component is held stationary.

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In one arrangement, the planetary carrier (green) is held stationary, and the sun gear

(yellow) is used as input. In this case, the planetary gears simply rotate about their own

axes (i.e., spin) at a rate determined by the number of teeth in each gear. If the sun

gear has Ns teeth, and each planet gear has Np teeth, then the ratio is equal to −Ns/Np.

For instance, if the sun gear has 24 teeth, and each planet has 16 teeth, then the ratio

is −24/16, or −3/2; this means that one clockwise turn of the sun gear produces

1.5counterclockwise turns of each of the planet gear(s) about its axis.

This rotation of the planet gears can in turn drive the annulus (not depicted in diagram),

in a corresponding ratio. If the annulus has Na teeth, then the annulus will rotate

by Np/Naturns for each turn of the planet gears. For instance, if the annulus has 64

teeth, and the planets 16, one clockwise turn of a planet gear results in 16/64, or 1/4

clockwise turns of the annulus. Extending this case from the one above:

One turn of the sun gear results in   turns of the planets

One turn of a planet gear results in   turns of the annulus

So, with the planetary carrier locked, one turn of the sun gear results in   turns

of the annulus.

The annulus may also be held fixed, with input provided to the planetary gear carrier;

output rotation is then produced from the sun gear. This configuration will produce an

increase in gear ratio, equal to 1+Na/Ns.[citation needed]

If the annulus is held stationary and the sun gear is used as the input, the planet carrier

will be the output. The gear ratio in this case will be 1/(1+Na/Ns). This is the lowest gear

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ratio attainable with an epicyclic gear train. This type of gearing is sometimes used

in tractors and construction equipment to provide high torque to the drive wheels.

In bicycle hub gears, the sun is usually stationary, being keyed to the axle or even

machined directly onto it. The planetary gear carrier is used as input. In this case the

gear ratio is simply given by (Ns+Na)/Na. The number of teeth in the planet gear is

irrelevant.

Compound planets of a Sturmey-Archer AM bicycle hub (gear ring removed)

FIXED CARRIER TRAIN RATIO

A convenient approach to determine the various speed ratios available in a planetary

gear train begins by considering the speed ratio of the gear train when the carrier is held

fixed. This is known as the fixed carrier train ratio.[2]

In the case of a simple planetary gear train formed by a carrier supporting a planet gear

engaged with a sun and annular gear, the fixed carrier train ratio is computed as the

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speed ratio of the gear train formed by the sun, planet and annular gears on the fixed

carrier. This is given by,

In this calculation the planet gear is an idler gear.

The fundamental formula of the planetary gear train with a rotating carrier is

obtained by recognizing that this formula remains true if the angular velocities of the

sun, planet and annular gears are computed relative to the carrier angular velocity.

This becomes,

This formula provides a simple way to determine the speed ratios for the simple

planetary gear train under different conditions:

1. The carrier is held fixed, ωc=0,

2. The annular gear is held fixed, ωa=0,

3. The sun gear is held fixed, ωs=0,

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Each of the speed ratios available to a simple planetary gear train can be obtained by

using band brakes to hold and release the carrier, sun or annular gears as needed. This

provides the basic structure for an automatic transmission.

SPUR GEAR DIFFERENTIAL

A spur gear differential constructed by engaging the planet gears of two co-axial

epicyclic gear trains. The casing is the carrier for this planetary gear train.

A spur gear differential is constructed from two identical coaxial epicyclic gear trains

assembled with a single carrier such that their planet gears are engaged. This forms a

planetary gear train with a fixed carrier train ratio R = -1.

In this case, the fundamental formula for the planetary gear train yields,

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or

Thus, the angular velocity of the carrier of a spur gear

differential is the average of the angular velocities of the sun

and annular gears.

In discussing the spur gear differential, the use of the term annular gear is a convenient

way to distinguish the sun gears of the two epicyclic gear trains. The second sun gear

serves the same purpose as the annular gear of a simple planetary gear train, but

clearly does not have the internal gear mate that is typical of an annular gear.[1]

§Gear ratio of Reversed Epicyclic Gearing

Some epicyclic gear trains employ two planetary gears which mesh with each other.

One of these planets meshes with the sun gear, the other planet meshes with the

annulus (or ring) gear. This results in different ratios being generated by the planetary.

The fundamental equation becomes:

where 

which results in:

when the carrier is locked,

when the sun is locked,

when the annulus is locked.

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"Compound planetary gear" is a general concept and it refers to any planetary

gears involving one or more of the following three types of structures: meshed-

planet (there are at least two more planets in mesh with each other in each planet

train), stepped-planet (there exists a shaft connection between two planets in each

planet train), and multi-stage structures (the system contains two or more planet sets).

Some designs use "stepped-planet" which have two differently-sized gears on either

end of a common casting. The large end engages the sun, while the small end

engages the annulus. This may be necessary to achieve smaller step changes in gear

ratio when the overall package size is limited. Compound planets have "timing marks"

(or "relative gear mesh phase" in technical term). The assembly conditions of

compound planetary gears are more restrictive than simple planetary gears,[8] and they

must be assembled in the correct initial orientation relative to each other, or their teeth

will not simultaneously engage the sun and annulus at opposite ends of the planet,

leading to very rough running and short life. Compound planetary gears can easily

achieve larger transmission ratio with equal or smaller volume. For example,

compound planets with teeth in a 2:1 ratio with a 50T annulus would give the same

effect as a 100T annulus, but with half the actual diameter.

More planet and sun gear units can be placed in series in the same annulus housing

(where the output shaft of the first stage becomes the input shaft of the next stage)

providing a larger (or smaller) gear ratio. This is the way some automatic

transmissions work.

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During World War II, a special variation of epicyclic gearing was developed for

portable radar gear, where a very high reduction ratio in a small package was needed.

This had two outer annular gears, each half the thickness of the other gears. One of

these two annular gears was held fixed and had one tooth fewer than did the other.

Therefore, several turns of the "sun" gear made the "planet" gears complete a single

revolution, which in turn made the rotating annular gear rotate by a single tooth. [citation

needed]

Planetary gear trains provide high power density in comparison to standard parallel axis

gear trains. They provide a reduction volume, multiple kinematic combinations, purely

torsional reactions, and coaxial shafting. Disadvantages include high bearing loads,

constant lubrication requirements, inaccessibility, and design complexity.[9][10]

The efficiency loss in a planetary gear train is 3% per stage. This type of efficiency

ensures that a high proportion of the energy being input is transmitted through the

gearbox, rather than being wasted on mechanical losses inside the gearbox.

The load in a planetary gear train is shared among multiple planets, therefore torque

capability is greatly increased. The more planets in the system, the greater the load

ability and the higher the torque density.

The planetary gear train also provides stability due to an even distribution of mass and

increased rotational stiffness. Torque applied radially onto the gears of a planetary gear

train is transferred radially by the gear, without lateral pressure on the gear teeth.

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SUN AND PLANET GEAR

The sun and planet gear (also called the planet and sun gear) was a method of

converting reciprocating motion to rotary motion and was used in the first rotative beam

engines.

It was invented by the Scottish engineer William Murdoch, an employee of Boulton and

Watt, but was patented by James Watt in October 1781. It was invented to bypass the

patent on the crank, already held by James Pickard.[1] It played an important part in the

development of devices for rotation in the Industrial Revolution.

The sun and planet gear converted the vertical motion of a beam, driven by a steam

engine, into circular motion using a 'planet', acogwheel fixed at the end of the

connecting rod (connected to the beam) of the engine. With the motion of the beam, this

revolved around, and turned, the 'sun', a second rotating cog fixed to the drive shaft,

thus generating rotary motion. An interesting feature of this arrangement, when

compared to that of a simple crank, is that when both sun and planet have the same

number of teeth, the drive shaft completes two revolutions for each double stroke of the

beam instead of one. The planet gear is fixed to the connecting rod and thus does not

rotate around its own axis.

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Note that the axle of the planet gear is tied to the axle of the sun gear by a link that

freely rotates around the axis of the sun gear and keeps the planet gear engaged with

the sun gear but does not contribute to the drive torque. This link appears, at first sight,

to be similar to a crank but the drive is not transmitted through it. Thus, it did not

contravene the crank patent.

The sun and planet gear converted the vertical motion of a beam, driven by a steam

engine, into circular motion using a 'planet', acogwheel fixed at the end of the

connecting rod (connected to the beam) of the engine. With the motion of the beam, this

revolved around, and turned, the 'sun', a second rotating cog fixed to the drive shaft,

thus generating rotary motion. An interesting feature of this arrangement, when

compared to that of a simple crank, is that when both sun and planet have the same

number of teeth, the drive shaft completes two revolutions for each double stroke of the

beam instead of one. The planet gear is fixed to the connecting rod and thus does not

rotate around its own axis.

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Note that the axle of the planet gear is tied to the axle of the sun gear by a link that

freely rotates around the axis of the sun gear and keeps the planet gear engaged with

the sun gear but does not contribute to the drive torque. This link appears, at first sight,

to be similar to a crank but the drive is not transmitted through it. Thus, it did not

contravene the crank patent.

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HARMONIC DRIVE

Harmonic Drive is a strain wave gear that can improve certain characteristics

compared to traditional gearing systems. Harmonic Drive is trademarked by the

Harmonic Drive company.

The strain wave gear was invented in 1957 by C.W. Musser. The advantages include:

no backlash, compactness and light weight, high gear ratios, reconfigurable ratios within

a standard housing, good resolution and excellent repeatability (linear representation)

when repositioning inertial loads,[1] high torque capability, and coaxial input and output

shafts.[2] High gear reduction ratios are possible in a small volume (a ratio from 30:1 up

to 320:1 is possible in the same space in which planetary gears typically only produce a

10:1 ratio).

They are typically used in industrial motion control, machine tool, printing

machine, robotics[3] and aerospace,[4] for gear reduction but may also be used to

increase rotational speed, or for differential gearing.

History

The basic concept of strain wave gearing (SWG) was introduced by C.W. Musser in his

1957 patent.[5] It was first used successfully in 1960 by USM Co. and later by Hasegawa

Gear Works, Ltd. under license of USM. Later, Hasegawa Gear Works, Ltd. became

Harmonic Drive Systems Inc. located in Japan and USM Co. Harmonic Drive division

became Harmonic Drive Technologies Inc.

The electrically-driven wheels of the Apollo Lunar Rover included strain wave gears in

their gearing. Also, the winches used on Skylab to deploy the solar panels were

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powered using strain wave gears. Both of these system were developed by The

Harmonic Drive Division of United Shoe Machinery Corp.[citation needed]

On January 1, 2006, Harmonic Drive Technologies/Nabtesco of Peabody, MA and HD

Systems of Hauppauge, NY, merged to form a new joint venture, Harmonic Drive LLC.

[2] HD Systems, Inc. was a subsidiary company of Harmonic Drive System, Inc. Offices

are maintained in Peabody, MA, Hauppauge, NY, San Jose, CA and Oak Park, IL.

Harmonic Drive Systems, Inc., Japan is headquartered in Tokyo with its primary

manufacturing location in Hotaka, Japan. Harmonic Drive AG has its European

headquarters and manufacturing in Limburg/Lahn,Germany.

Mechanics

The strain wave gearing theory is based on elastic dynamics and utilizes the flexibility of

metal. The mechanism has three basic components: a wave generator, a flex spline,

and a circular spline. More complex versions have a fourth component normally used to

shorten the overall length or to increase the gear reduction within a smaller diameter,

but still follow the same basic principles.

The wave generator is made up of two separate parts: an elliptical disk called a wave

generator plug and an outer ball bearing. The gear plug is inserted into the bearing,

giving the bearing an elliptical shape as well.

The flex spline is like a shallow cup. The sides of the spline are very thin, but the bottom

is thick and rigid. This results in significant flexibility of the walls at the open end due to

the thin wall, but in the closed side being quite rigid and able to be tightly secured (to a

shaft, for example). Teeth are positioned radially around the outside of the flex spline.

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The flex spline fits tightly over the wave generator, so that when the wave generator

plug is rotated, the flex spline deforms to the shape of a rotating ellipse but does not

rotate with the wave generator.

The circular spline is a rigid circular ring with teeth on the inside. The flex spline and

wave generator are placed inside the circular spline, meshing the teeth of the flex spline

and the circular spline. Because the flex spline has an elliptical shape, its teeth only

actually mesh with the teeth of the circular spline in two regions on opposite sides of the

flex spline, along the major axis of the ellipse.

Assume that the wave generator is the input rotation. As the wave generator plug

rotates, the flex spline teeth which are meshed with those of the circular spline change.

The major axis of the flex spline actually rotates with wave generator, so the points

where the teeth mesh revolve around the center point at the same rate as the wave

generator. The key to the design of the strain wave gear is that there are fewer teeth

(for example two fewer) on the flex spline than there are on the circular spline. This

means that for every full rotation of the wave generator, the flex spline would be

required to rotate a slight amount (two teeth, for example) backward relative to the

circular spline. Thus the rotation action of the wave generator results in a much slower

rotation of the flex spline in the opposite direction.

For a strain wave gearing mechanism, the gearing reduction ratio can be calculated

from the number of teeth on each gear:

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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 either 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.

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NOMENCLATURE

General nomenclature

Rotational frequency, n

Measured in rotation over time, such as RPM.

Angular frequency, ω

Measured in radians/second.   rad/second

Number of teeth, N

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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

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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 basic dimension

and cannot be measured, but is a location where other measurements are made. Its

value is based on the number of teeth, the normal module (or normal diametral pitch),

and the helix angle. It is calculated as:

in metric units or   in imperial units.[22]

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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, z1 and z2 are the number of cogs (teeth) for each of the two

wheels (gears). These numbers (or at least one of them) is often chosen

amongprimes 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

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rotational direction while maintaining uniform wear with all 4 gears in this case.

Mechanic engineers at least in continental Europe 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.[23]

Operating pitch diameters

Diameters determined from the number of teeth and the center distance at which gears

operate.[7] 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, 

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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

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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, where the tooth profile is 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

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Interference

Contact between teeth other than at the intended parts of their surfaces.

Interchangeable setA set of gears, any of which mates properly with any other

HELICAL GEAR NOMENCLATURE

Helix angle, 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, pCircular 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 NOMENCLATURE

LeadDistance 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, pDistance 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.

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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 NOMENCLATURE

Line of contact

Path of action

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Line of action

Plane of action

Lines of contact (helical gear)

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Arc of action

Length of action

Limit diameter

90 | P a g e

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

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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

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The distance on the line of action through which the point of contact moves during the

action of the tooth profile.

Arc of action, Qt

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, Qa

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, Qr

The arc of the pitch circle through which a tooth profile moves from contact at the pitch

point until contact ends.

Contact ratio, mc, 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 in which a tooth comes and goes out of

contact with the mating gear.

Transverse contact ratio, mp, εα

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.

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Face contact ratio, mF, εβ

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, mt, εγ

The sum of the transverse contact ratio and the face contact ratio.

Modified contact ratio, mo

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

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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 NOMENCLATURE

Tooth thickness

Thickness relationships

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Chordal thickness

Tooth thickness measurement over pins

Span measurement

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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

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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

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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.

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PITCH NOMENCLATURE

For other uses, see Pitch.

Pitch is the distance between a point on one tooth and the corresponding point on an

adjacent tooth.[7] 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

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Tooth pitch

Base pitch relationships

Principal pitches

Circular pitch, p

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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

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In an involute gear, the pitch 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 diametral pitch, Pnd

Value of diametral pitch in a normal plane of a helical gear or worm.

Angular pitch, θN, τ

Angle subtended by the circular pitch, usually expressed in radians.

degrees or   radians

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BACKLASH

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 in which precision is important, such as instrumentation and control,

backlash can be minimised 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

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achieves, in effect, a single gear with expanding teeth. Another method involves

tapering the teeth in the axial direction and providing for the gear to be slid in the axial

direction to take up slack.

Shifting of gears

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.[24]

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TOOTH PROFILE

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

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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

provides a constant velocity ratio. 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 have been by far the most commonly

used in modern times. They are 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.

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GEAR MATERIALS

Wooden gears of a historicwindmill

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[25] and the ability to be made with materials not needing additional

lubrication. Manufacturers have employed plastic gears to reduce costs in consumer

items including copy machines, optical storage devices, cheap dynamos, consumer

audio equipment, servo motors, and printers.

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§Standard pitches and the module system[edit]

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 inchmeasurements, standard diametral pitch values with units

of "per inch" are chosen; the diametral 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.[26][27] 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"[27]

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 millimeters

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 diametral 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.[28] 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.[29]

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MANUFACTURE

Gear Cutting simulation (length 1m35s) faster, high bitrate version.

As of 2014, an estimated 80% of all gearing produced worldwide is produced by net

shape molding. Molded gearing is usually eitherpowder metallurgy or plastic.[30] 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.

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Inspection

Overall gear geometry can be inspected and verified using various methods such

as industrial CT scanning, coordinate-measuring machines, white light scanner or laser

scanning. Particularly useful for plastic gears, industrial CT scanning can inspect

internal geometry and imperfections such as porosity.

Important dimensional variations of gears result from variations in the combinations of

the dimensions of the tools used to manufacture them. An important parameter for

meshing qualities such as backlash and noise generation is the variation of the actual

contact point as the gear rotates, or the instantaneous pitch radius. Precision gears

were frequently inspected by a method that produced a paper "gear tape" record

showing variations with a resolution of .0001 inches as the gear was rotated.[27]

The American Gear Manufacturers Association was organized in 1916 to formulate

quality standards for gear inspection to reduce noise from automotive timing gears; [31] in

1993 AGMA assumed leadership of the ISO committee governing international

standards for gearing. The ANSI/AGMA 2000 A88 Gear Classification and Inspection

Handbook specifies quality numbers from Q3 to Q15 to represent the accuracy of tooth

geometry; the higher the number the better the tolerance. [32] Some dimensions can be

measured to millionths of an inch in controlled-environment rooms.

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GEAR MODEL IN MODERN PHYSICS

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 electrical current as a rotation of

particles in opposite directions to that of the rotating field lines.[33]

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.[34]

The Three Wave Hypothesis compares the wave–particle duality to a bevel gear.[35]

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