### Formulae for gears

In the imperial system everything is measured in inches. In the metric system everything is measured in millimeters. Any gear made using either of these systems can only be meshed properly with other gears made to the same system.

#### Formulae for spur gears

The key parameter for any spur gear is the pitch circle diameter whether it is metric or imperial.

PCD     pitch circle diameter – this is the effective diameter of the gear. If two gears are meshed together then the distance between their centers will be –

(PCD1 + PCD2)/2

The next important parameter is the size of the teeth

In the metric system this is called Module or mod.

mod = PCD / number of teeth

in the imperial system this is the diametrical pitch or DP.

DP = number of teeth / PCD

Notice that a large mod number is a large tooth size whereas a large DP number is a small tooth size.

It is sometimes useful to be able to compare metric sizes with imperial sizes. The relationship is:

mod = 25.4 / DP

or

DP = 25.4 / mod

Spur gears – example

Suppose we need to make a gear where the size of teeth is 1 mod and the number of teeth is n. We need to know the outside diameter, OD. This is;

OD = ( N + 2 ) * mod

The pitch circle diameter, PCD,  will be:

PCD = n * mod

The outside diameter of the gear is OD

In the metric system OD is –

OD = (number of teeth + 2) * MOD

In the imperial system OD is

OD = (number of teeth + 2) / DP

Circular pitch is the space between one tooth and the next on the  pitch circle.

In the metric system this is

CP =  Mod * pi

In the imperial system this is

CP = pi / DP

The addendum is the height of the tooth above the pitch circle

In the metric system this is –

In the imperial system this is –

A module size can be converted into dp and vice versa

DP = 25.4 / Module

#### Spur gears and pressure angles

It might be noticed that none of the above involve the pressure angle. This does not mean that the pressure angle has no effect. In the same way that the outside diameter touches the top of all of the teeth there is also a base circle that touches the bottoms of the teeth. The pitch circle is somewhere in between. If the addendum is the space between the pitch circle and the outside diameter then the space between the pitch circle and the base circle is the dedendum.

When cutting gears the depth of cut marked on the cutter is the height of the tooth plus an allowance to be sure of clearance. So long as the outside diameter is right then the depth marked on the cutter will be right.

The base circle is:

Base circle (BC) = PCD * cos (pressure angle)

Then:

Dedendum = PCD – BC

or

Dedendum = PCD – PCD * cos (pressure angle)

These formulae are the same for both the metric and imperial systems though the units used would be different.

#### Formulae for helical gears

The formulae for helical gears are the same as for spur gears except for the the PCD and the number of teeth and a completely new one for determining the “lead”.

On a spur gear all the teeth form straight flutes. On a helical gear all the teeth are each in the form of a helix. The angle of this helix is the helix angle –  α.

The helix is usually formed by driving the auxiliary input on the dividing head from the leadscrew of the milling table. The result is that as the workpiece moves past the cutter it slowly rotates.

When making a helical gear we will usually start knowing the helix angle, the size and the number of teeth the gear must have. The key calculation is determining the “lead”. The lead is the distance the workpiece moves along the table whilst making one complete revolution. The lead is related to the PCD of the gear and the helix angle by  the following formula –

lead = (PCD * pi) / tan α

On a helical gear the PCD is dependent on the helix angle

In metric the PCD is:

PCD = (number of teeth * mod) / cos α

In imperial the PCD is:

PCD = (no of teeth) /  (DP * cos (α))

The number of teeth is:

In metric

number of teeth = PCD * cos (α)  / mod

In imperial

number of teeth = DP * cos (α) * PCD