**Gears** are machine elements that transmit motion by
means of successively engaging teeth. The gear teeth act like small levers.

Gears may be classified according to the relative position of the axes of revolution. The axes may be

- parallel,
- intersecting,
- neither parallel nor intersecting.

**Gears for connecting parallel shafts**

*Spur gears*The left pair of gears makes

**external contact**, and the right pair of gears makes**internal contact***Parallel helical gears**Herringbone gears*(or double-helical gears)*Rack*and*pinion*(The rack is like a gear whose axis is at infinity.)

**Gears for connecting intersecting
shafts**

*Straight bevel gears**Spiral bevel gears*

**Neither parallel nor intersecting
shafts**

*Crossed-helical gears**Hypoid gears**Worm and wormgear*

Figure 1 shows two mating gear teeth,

Although the two profiles have different velocities *
V_{1}* and

Point *P* is very important to the velocity ratio,
and it is called the **pitch point**. Pitch point divides the line between
the line of centers and its position decides the velocity ratio of the two
teeth. The above expression is the **fundamental law of gear-tooth action**.

For a constant velocity ratio, the position of *P*
should remain unchanged. In this case, the motion transmission between two gears
is equivalent to the motion transmission between two imagined slipless cylinders
with radius *R _{1}* and

The** fundamental law of gear-tooth action** may now
also be stated as follow (for gears with fixed center distance):

The common normal to the tooth profiles at the point of contact must always pass through a fixed point (the pitch point) on the line of centers (to get a constant velocity ration).

To obtain the expected *velocity ratio* of two tooth
profiles, the normal line of their profiles must pass through the corresponding
pitch point, which is decided by the *velocity ratio*. The two profiles
which satisfy this requirement are called **conjugate profiles**. Sometimes,
we simply termed the tooth profiles which satisfy the *fundamental law of
gear-tooth action* the *conjugate profiles*.

Although many tooth shapes are possible for which a
mating tooth could be designed to satisfy the fundamental law, only two are in
general use: the *cycloidal* and *involute* profiles. The involute has
important advantages -- it is easy to manufacture and the center distance
between a pair of involute gears can be varied without changing the velocity
ratio. Thus close tolerances between shaft locations are not required when using
the involute profile. The most commonly used *conjugate* tooth curve is the
*involute curve*

The following examples are involute spur gears. We use
the word *involute* because the contour of gear teeth curves inward. Gears
have many terminologies, parameters and principles. One of the important
concepts is the *velocity ratio,* which is the ratio of the rotary velocity
of the driver gear to that of the driven gears.

The curve most commonly used for gear-tooth profiles is
the involute of a circle. This **involute curve** is the path traced by a
point on a line as the line rolls without slipping on the circumference of a
circle. It may also be defined as a path traced by the end of a string which is
originally wrapped on a circle when the string is unwrapped from the circle. The
circle from which the involute is derived is called the **base circle**.

Figure 3 shows some of the terms for gears.

In the following section, we define many of the terms used in the analysis of spur gears. Some of the terminology has been defined previously but we include them here for completeness.

**Pitch surface**: The surface of the imaginary rolling cylinder (cone, etc.) that the toothed gear may be considered to replace.**Pitch circle**: A right section of the pitch surface.**Addendum circle**: A circle bounding the ends of the teeth, in a right section of the gear.**Root (or dedendum) circle**: The circle bounding the spaces between the teeth, in a right section of the gear.**Addendum**: The radial distance between the pitch circle and the addendum circle.**Dedendum**: The radial distance between the pitch circle and the root circle.**Clearance**: The difference between the dedendum of one gear and the addendum of the mating gear.**Face of a tooth**: That part of the tooth surface lying outside the pitch surface.**Flank of a tooth**: The part of the tooth surface lying inside the pitch surface.**Circular thickness**(also called the**tooth thickness**) : The thickness of the tooth measured on the pitch circle. It is the length of an arc and not the length of a straight line.**Tooth space**: The distance between adjacent teeth measured on the pitch circle.**Backlash**: The difference between the circle thickness of one gear and the tooth space of the mating gear.**Circular pitch**p: The width of a tooth and a space, measured on the pitch circle.**Diametral pitch**P: The number of teeth of a gear per inch of its pitch diameter. A toothed gear must have an integral number of teeth. The*circular pitch*, therefore, equals the pitch circumference divided by the number of teeth. The*diametral pitch*is, by definition, the number of teeth divided by the*pitch diameter*. That is,and

Hence

where

- p = circular pitch
- P = diametral pitch
- N = number of teeth
- D = pitch diameter

That is, the product of the diametral pitch and the circular pitch equals .

**Module**m: Pitch diameter divided by number of teeth. The pitch diameter is usually specified in inches or millimeters; in the former case the module is the inverse of diametral pitch.**Fillet**: The small radius that connects the profile of a tooth to the root circle.**Pinion**: The smaller of any pair of mating gears. The larger of the pair is called simply the gear.**Velocity ratio**: The ratio of the number of revolutions of the driving (or input) gear to the number of revolutions of the driven (or output) gear, in a unit of time.**Pitch point**: The point of tangency of the pitch circles of a pair of mating gears.**Common tangent**: The line tangent to the pitch circle at the pitch point.**Line of action**: A line normal to a pair of mating tooth profiles at their point of contact.**Path of contact**: The path traced by the contact point of a pair of tooth profiles.**Pressure angle**: The angle between the common normal at the point of tooth contact and the common tangent to the pitch circles. It is also the angle between the line of action and the common tangent.**Base circle**:An imaginary circle used in involute gearing to generate the involutes that form the tooth profiles.

Table lists the commonly used diametral pitches.

Coarse pitch | 2 | 2.25 | 2.5 | 3 | 4 | 6 | 8 | 10 | 12 | 16 |

Fine pitch | 20 | 24 | 32 | 40 | 48 | 64 | 96 | 120 | 150 | 200 |

Figure 4 shows two meshing gears contacting at point *K _{1}*
and

To get a correct meshing, the distance of *K _{1}K_{2}*
on gear 1 should be the same as the distance of