# Gears

A gears is a mechanical system composed of two or several toothed wheels being used with the Transmission of the movement as rotation. These two toothed wheels are in contact one with the other and are transmitted power by obstacle. Gears are composed of a pinion (thus one names the only wheel or the smallest wheel) and of a wheel, a toothed rack or a crown. When there is more than two toothed wheels, one speaks about spur gearing.

The most widespread profile, in general mechanics, is the profile in involute of circle.

There exist several types of teeth: right teeth and helical teeth (of propeller).

There exist several types of gears: nonconvergent axis, convergent axis, non-crossed gears (to which the gears with wheel and Endless screw belong) and pinion gears and Crémaillère.

## General information

The gears are used in all the branches of mechanics to transmit movements, clock industry to the reducers of heavy industry. The transmission is done with a very good energetic efficiency (>95% on gears under correct conditions of assembly). The speed ratio obtained between the entry and the exit depends only on the numbers of teeth of the wheels in contact.

For transmissions with great distance between centres, compared to the dimension of the parts, one will prefer a chain, a Courroie or a cascade of gears.

### Vocabulary

The terms according to are employed in the continuation of the article.
• Teeth: toothed part of a machine element.
• Profile: it is the form, in a cross-section, side of a tooth.
• Module: generating dimensional parameter relating to the periodicity of the teeth thus to their size.
• Gears: together of two or several machine elements comprising of teeth and intended to gear together.
• Gears with Axis S parallels: gears whose axes are parallel.
• convergent Gears: gears whose axes have a point of intersection.
• Skew type gears: gears whose axes are neither parallel, nor convergent.
• Ratio transmission ( R ): report/ratio velocity emission on the speed of entry, is also many teeth of the entry - known as driving - on the number of teeth of the exit - known as carried out - gears. If R is higher than 1 one speaks about multiplier, if R is lower than 1 one speaks about reducer.

## Teeth

There exist several types of teeth, with the particular properties. Almost the whole of the forms known as are combined: during rotation, the teeth remain in contact in a sagittal plan, and when the locus of this contact point is a line, the tooth shapes are involutes of circle. A notable exception is the Novikov gears, known as also sometimes of Fisher, in which the contact between two teeth is done during a “specific” time all along the profile. These teeth are thus always helicoid, make it possible to transmit important powers with very good outputs, even if the pinion has only few teeth, but require a rigorous positioning.

### Profile in involute of circle

It is the profile almost universally used for the transmission of power.

The Développante of the circle is the trajectory of a point of a right which rolls without slipping on a Cercle. This circle is called basic circle , of diameter db. The zone of existence of the involute is between the basic circle and the Infini. There does not exist involute inside the basic circle. One thus should not seek to make function gears inside the basic circles of teeth who constitute it.

If two basic circles are considered, associated with two wheels with the same gears, it is possible to make roll without simultaneously slipping a line on the two circles. Of this fact the circumferential Speed of the points of the circles is the same one as those of the right-hand side. A point of the right-hand side (not of engaging) will generate, on the two pinions, the blank of tooth.

If the line passes between the centers of the circles one obtains the traditional gears, the wheels turn then in contrary direction, and the Rapport of transmission depends on the diameters. When it is external, the gears are known as paradoxical and the wheels turn in the same direction.

In the case of the traditional gears, and more particularly of the standard gears, the basic circles are brought closer so that the interior line forms a angle of pressure α with the line which passes by the axes. According to the standard α is worth 20° in Europe, 25° with US and 12.5° for the old gears. The teeth are limited to a zone around item I, said not engaging, where speeds of slip of the teeth are negligible, which contributes to a optimal Rendement of the gears. One obtains the two flanks of tooth by considering the two interior tangents. The force exerted of a tooth on the other breaks up into 2: tangeantielle (useful) which transmits the couple, and radial (parasite) which tend to move away the wheels. A small angle of pressure to the advantage of limiting this force of parasitic repulsion, but gives the fragile shape of tooth. on the other hand a high angle of pressure gives squat teeth thus more resistant, but generates many forces on the axes.

The paradoxical gears are used in certain Différentiel S (differential Mercier engineer Renault). Important speeds of relative slip on the teeth allow a “blocking” partial of the differential when the wheels of the vehicle do not have the same grip the ground. It is not a question of blocking strictly speaking since resistance to the movement is not obtained by obstacle if not Frottement. To ensure the relay of the catch of the teeth, it is often necessary to lay out the teeth in different radial plans, or to have recourse to a tooth in Hélice (continuous solution).

See here a video on the paradoxical gears.

### Generation of the teeth

#### Right teeth

The generator of form of the teeth is a straight line parallel with the axis of rotation. It is the type of teeth more running. It is used in all the applications of general mechanics. It is this system which makes it possible to transmit the maximum of effort. Its principal defect is to be noisy.

#### Helical teeth

The generator of form of the teeth is a helicoid line of the same axis than the axis of rotation. This type of teeth has the advantage of being quieter than right teeth, by creating less vibrations. The helical teeth also make it possible to increase the control of the transmission, by making so that the number of teeth simultaneously in contact becomes constant, which makes it possible to transmit greater efforts and especially to attenuate the vibrations and the noises. N the other hand this type of teeth generates an axial load whose intensity depends on the angle of inclination of teeth. The bearings or the stages must be dimensioned to take again this effort.

For the non-crossed gears, the propellers are obligatorily of contrary direction so that teeth can gear, except in the very particular case of the paradoxical gears.

#### Double helical gearings

Herring-bone teeth, or teeth “Citroen” (from where its logo), is made up of two helical teeth settings in opposition so as to cancel the axial load. Although alluring from the theoretical point of view, in practice this type of teeth is complicated, therefore expensive to realize. Herring-bone teeth are used only in heavy industry, most of the time they are two gears (with contrary propellers) combined and not the pinions monoblocs.

#### Screw gears

A gears with screw is skew type gears made up of a screw and a combined worm wheel. The profile of the screw is (in general) trapezoidal.

In many cases this device is irreversible, which means that if the screw can involve the wheel, the wheel cannot, because them frictions, to actuate the screw. This case is interesting for example for the ordering of a winch which cannot be held all alone.

## Geometrical study

### For the non-crossed gears

The formulas below are valid for standardized teeth.
• diameters:
$d_i = Mr. z_i$
• distance between centres:

$a = \ frac \left\{\left(d_1 + d_2\right)\right\}\left\{2\right\} \, = \ frac \left\{m \left(z_1 + z_2\right)\right\}\left\{2\right\} \,$
• gear ratio ($z_1$: pinion, $z_2$: coil):

$u = \ frac \left\{z_2\right\} \left\{z_1\right\} \,$
• ratio transmission (speeds) since a tree of entry (E) towards an output shaft (S) through 1 external gear:

$i = \ frac \left\{\ omega_ \ mathrm \left\{S\right\}\right\} \left\{\ omega_ \ mathrm \left\{E\right\}\right\} = \ frac \left\{N_ \ mathrm \left\{S\right\}\right\} \left\{N_ \ mathrm \left\{E\right\}\right\} = \left(- 1\right) \ frac \left\{z_ \ mathrm \left\{E\right\}\right\} \left\{z_ \ mathrm \left\{S\right\}\right\} \,$

with:

• Z: the number of teeth,
• NR: ($\ Omega \,$): the rotation expressed in tr/min (rad/s),
• C: couples with the wheel expressed in N.m,
If the speed of entry is higher than the velocity emission, the report/ratio is lower than 1: it is a reducer.
• ratio transmission of a spur gearing:

$i_ \ mathrm \left\{early\right\} = i_ \ mathrm \left\{I\right\} \ cdot i_ \ mathrm \left\{II\right\} \ cdot i_ \ mathrm \left\{III\right\} \ ldots \ cdot i_ \ mathrm \left\{N\right\} \,$

### Cutting of the pinions

Teeth are carried out mainly by removal of matter (machining). It is generally about an engaging simulated between a tool (pinion, toothed rack, or mill) and the wheel to be cut. So the module of teeth is imposed by the tools.

The modules are standardized. There are the principal values, the secondary values (between brackets) and the allowed values in exceptional circumstances (between brackets and in Italic ):

d=m.z

### Conditions of engaging

It is not possible to produce any gears. The good conditions of engaging limit the choice of the number of teeth of each pinion. The criteria to be considered are:
• Interférence enters the teeth:
• Distribution of wears.
The numbers of teeth must be if possible selected first between them (what makes it possible each tooth of a wheel to meet all the teeth of the other).
• Report/ratio of control.

It is necessary to optimize the number of teeth in catch for better distributing the loads, and thus at the same time to decrease the effects of tiredness on the teeth, and to reduce the noise.
• Offset of teeth and modification of distance between centres

A couple of pinions given can function since teeth are sufficiently overlapping. Even if there is play, the distance between centres being then larger. In this case, it is possible to cancel the play by inflating the teeth of two pinions (what amounts reducing the projection to the profit of the tooth). The transmission ratio is unchanged but the diameters are modified.

The majority of the standard gears are without offset of teeth (the tooth being then as large as the projection), but in very pointed cases (gear box) that is practiced for two reasons mainly:

• there does not exist couple (Z1, Z2) making it possible to ensure at the same time report/ratio (Z1/Z2) and distance between centres (Z1+Z2), it is thus necessary to vary (artificially) its value by off-setting at least teeth.
• the teeth of the small pinion, more often solicited, are enlarged, and those of the large pinion reduced, in order to confer to them the same lifespan (concept of tiredness).

In general, if there is offset of teeth, it is not always possible to change one pinion in gears.

## Types of gears

As each wheel (with involute of circle) can be regarded as having been generated by a basic rack, the first type is the gears with wheel and toothed rack.

### Parallel or cylindrical gears

The axes of the two toothed wheels are parallel.

### Convergent or conical gears

The axes of the two toothed wheels are convergent.

### Skew type gears

The axes of the two toothed wheels are not in the same plan. For example:
• helicoid Skew type gears
• Hypoid gears (not to be confused with the bevel gear)
• Wheel and screw with restricted gears

## Spur gearings

A spur gearing is a combination of gears.

### Simple train

The transmission ratio is the product of the numbers of teeth of the driving wheels divided by that of the driven gears.

### Planetary gear (or epicycloidal)

They are systems made up of satellite S assembled on a satellite carrier turning around two planet gears. They thus present three variable components compared to another fixed. They are used such as they are in the systems Différentiel S.

By blocking an element, one obtains, with the same geometry, various reports/ratios of reduction between the still mobile elements. It is besides the principle used in the “automatic” gear boxes.

These trains are very much used in mechanics because they can provide enormous reports/ratios of reduction, with parts of reasonable size, and acceptable outputs. Moreover their geometry often leads to a configuration where the tree of entry is coaxial with the output shaft. One easily finds reducers epicycloidal in the trade compatible with electrical motors (becoming blow motor reducer).

### Spherical train

On the principle the spherical train approaches the epicyclic gear. The gears are conical and thus seem laid out on a sphere. It is the geometry of the differential of the driving axles of the motor vehicles. They easily combine the function bevel gearbox, the reduction, and the differential function.

## Deterioration of teeth

Teeth can be deteriorated in two manners:
• by rupture of one or more teeth,
• by wear of surfaces of contact. This last item is discussed in detail in one of the chapters of the wikilivre of tribology devoted to the damage of teeth.

## External bonds

• Profiles of combined gears: mathematical curves of profiles of teeth.