**Introduction.**
** **When mating gear teeth are designed to produce a constant angular velocity
ratio during meshing they are said to have 'conjugate action'. To provide this an 'involute'
type profile is almost universally used for tooth forms.

**An example of extreme gear ratios:**
**Watch Gears:** IWC watch Portuguese Perpetual Calendar

The eternal moonphase display is so accurate that it will only be 1 day out of
sync with the moon itself after 577 years.

The reduction ratio between the seconds hand and the century slide is 6,315,840,000:1

A point on the balance will cover 1.6 million km in a year.

After 25,228,800,000 beats the century slide will move through 26^{o}
at the end of a century.

The IWC engineers have programmed the mechanical calendar until the year 2499.

**Design**

There are two modes that are important causes of gear failures. Bending stresses
(leading to tooth breakage) which are a maximum at the tooth root and compressive stresses
(leading to pitting) that are a maximum on the tooth face. Because tooth loading is cyclic,
both of these mechanisms are of a fatigue nature. The design of gears needs to counter both
of these potential failure modes. An important part of providing the resistance to the high
contact stresses is to use gears of appropriate hardness. The lower the levels of impurities
in a material, the better it is normally able to resist fatigue.

Link to page showing some gear nomenclature.

Quite a bit of information is available about gears on the following sites:

DR Gears site.

Quality Transmission Components site, go to QTC Technical Library.

**Loads on Teeth**

The tangential force on the teeth can be found from:

W_{t} = 60H/(3.14159.d.n) where:

W_{t} = transmitted load, N

H = power, W

d = gear diameter, m

n = speed, rev/min.

**Bending Stresses - The Lewis Formula**

Although this was published in 1893, it is still very widely used for assessing
bending stresses when designing gears. The method involves moving the tangential force and
applying it to the tooth tip and assuming the load is uniformly distributed accross the
tooth width with the tooth acting as a simple cantilever of constant rectangular cross
section, the beam depth being put equal to the thickness of the tooth root (t) and
the beam width being put equal to the tooth, or gear, width (b_{w}).

The section modulus is I/c = b_{w}t^{2}/6 so the bending
stress is given by:

sigma_{bending} = M/(I/c) = 6W_{t}L/(b_{w}t^{2}) eqn.1.

Assuming that the maximum bending stress is at point 'a'.

By similar triangles: or eqn.2

Rearranging eqn.1 gives:

Substitute the value for 'x' from eqn.2 and multiply the numerator and denominator by the circular pitch, 'p' gives:

let y=2x/3p then

This is the original Lewis equation and 'y' is called the Lewis form factor which may be determined graphically or by computation.

Engineers often now work with the 'diametral pitch', 'P', and or the 'module', 'm', which is 1/diametral pitch = 1/P

Then where Y = 2xP/3

Written in terms of the module:

The Lewis form factor considers only static loading, it is dimensionless, independent of tooth size and is a only a function of tooth shape. It does not take into account the stress concentration that exists in the tooth fillet.

The Lewis formula is generally limited to pitch line velocities up to 7.6 m/s and
based on tests (in the 19th Century) on cast iron gears with cast teeth, C G Barth
suggested a modification involving a velocity factor, K_{V}.

For cut or milled teeth the Barth equation (in SI units) is often modified to:

sigma_{bending, allowable} = W_{t}/b_{w}mYK_{V}

**NB Recent Changes in Barth Equation**

In about 2000 the AGMA (see below) re-defined the dynamic factor, K_{V},
as the inverse of that originally proposed by Barth, above. Consequently it is
greater than 1 (and called K_{V}' here) and the expression for the allowable
bending stress becomes:

sigma_{bending, allowable} = W_{t}K_{V}'/b_{w}mY

For a full depth tooth with a 20^{o} pressure angle, Y varies between
0.245 for a gear with 12 teeth to 0.471 for a gear with 300 teeth. (0.485 for a rack).

It is common for spur gears to be designed with a face width of between 3 and 5 times the circular pitch.

**American Gear Manufacturers Association (AGMA) Code**

Some key points from the AGMA approach to gear design are shown below.

AGMA have published graphs of allowable bending stresses and allowable surface contact stresses as a function of the Brinell hardness for some grades of through hardened steel.

Grade 2 steels, that have higher allowable stresses, are more closely specified than grade 1 steels.

This is commonly used and contains further refinements compared to the approach above,
it also includes detailed guidance about materials.

A number of modifying factors are normally included in the AGMA code:

K_{a} = application factor - depends on the type of power source

K_{s} = size factor - increases above 1 for a module, m, of
6 mm or greater.

K_{m} = load distribution factor - depends mainly on face width.

K_{v} = dynamic factor - depending upon tooth accuracy, loads greater
than the transmitted load may be generated.

**Contact Stresses**

These are determined by Hertzian contact stress analysis. The maximum pressure in the (rectangular) contact zone when two parallel cylinders are pressed together is given by:

where E' is the effective modulus of elasticity:

W' is the dimensionless
load = w'/(E'/R_{x})

w' is the load per unit width = normal load/gear width and

1/R_{x} = 2((1/pinion dia)+(1/gear dia))/sin(pressure angle)

The same modifying factors are again used with the contact stresses that were used with the bending stresses.

sigma_{compressive} = p_{H}(K_{a }K_{s} K_{m}/K_{v})^{0.5}

^{}

Link to example gear tooth strength calculation, note file size: 214 kB.

**Forces on Helical Gear Teeth**

As the teeth on helical gears are inclined to the axis of the gear, the tooth force generates
an axial or thrust load in addition to the radial force and the tangential force (which is the
only one that does useful work). The tooth load can be resolved in the three directions as shown
in this diagram.

The forces and resulting bending moments on the shaft carrying a helical gear are illustrated
in this note.

**Stresses in helical gear teeth**

When calculating the stresses in helical gear teeth, the Lewis form factor for the 'virtual number
of teeth' needs to be used rather than that for the actual number of teeth. This is because on looking along
a tooth on a helical gear the apparent radius of the gear is greater than that of the gear blank (the cross
section is an ellipse). The 'virtual number of teeth' is found by:

**Supplementary note on Stresses in Gears**

It should be noted that using the simplified version of these methods (including only K_{v})
on gears in automotive gear boxes gives high stresses, particularly for 1st gear. Gears for
car gear boxes
are probably manufactured to a high degree of accuracy, to keep noise levels low. The velocity
correction factor, K_{v}, from the Barth equation, is over 100 years old and probably gives a
conservative factor compared to that appropriate for modern high quality gears. Using all the
AGMA factors - and noting that 'Y' the Lewis form factor is NOT used but the geometry factors J or I,
(for bending and compressive stresses) are included, should give a more useful answer.

Note that 1st gear may be designed for a limited life as it so rarely operates under
maximum load.

Link to example on helical gear tooth strength calculation, 85kB file size.

**Torque Acting on a Gearbox**

As the input and output torques associated with a gearbox are normally not the same, some
'holding' torque will be needed to prevent the gearbox rotating. This can be determined as shown in
this diagram

**Lubrication**

During operation the teeth are sliding against one another, so to prevent wear
lubrication is normally essential for heavily loaded gears. Even though a gearbox may
have an efficiency of 97%, where considerable power is being transmitted, 3% loss as heat
generated within the gearbox, may necessitate the provision of some type of forced cooling.

**Acceleration of a Geared System**

- link to notes.
- example on a geared hoist, 56 kB file size

More detailed information about gears and some helpful animations can be seen in the chapter on 'Gears' at Mechanical Engineering Department pages at the University of Western Australia.

Useful information (albeit in imperial units) can be found on the 'Boston Gear' web site: click here

Further Reading: Shigley and Mischke, chapters 13, 14 and 15.

David J Grieve, Updated: 26th October 2010, Original: 24th November 2005.