# Notes on Spur Gears

## Definitions

The radial distance between the Pitch Circle and the top of the teeth.
Arc of Action:
That arc of the Pitch Circle between the first point of contact between gear teeth and the last.
Arc of Approach:
That arc of the Pitch Circle between the first point of contact between gear teeth and the the Pitch Point.
Arc of Recession:
That arc of the Pitch Circle between the Pitch Point and the last point of contact between gear teeth.
Backlash:
Play between mating teeth.
Base Circle:
The circle from which is generated the involute curve upon which the tooth profile is based.
Center Distance:
The distance between centers of two gears.
The distance between a chord, passing through the points where the Pitch Circle crosses the tooth profile, and the tooth top.
Chordal Thickness:
The thickness of the tooth measured along a chord passing through the points where the Pitch Circle crosses the tooth profile.
Circular Pitch:
Inches of Pitch Circle circumference per tooth.
Circular Thickness:
The thickness of the tooth measured along an arc following the Pitch Circle
Clearance:
The distance between the top of a tooth and the bottom of the space into which it fits on the meshing gear.
Contact Ratio:
The ratio of the length of the Arc of Action to the Circular Pitch.
Dedendum:
The radial distance between the bottom of the space between teeth and the top of the teeth.
Diametral Pitch:
Teeth per inch of diameter. Sometimes written (incorrectly) as Diametrical Pitch.
Face:
The working surface of a gear tooth, located between the pitch diameter and the top of the tooth.
Face Width:
The width of the tooth measured parallel to the gear axis.
Flank:
The working surface of a gear tooth, located between the pitch diameter and the bottom of the space between gear teeth
Gear:
The larger of two meshed gears. If both gears are the same size, they're both called "gears". See Pinion.
Land:
The top of the tooth.
Line of Action:
That line along which the point of contact between gear teeth travels, between the first point of contact and the last.
Module:
Teeth per millimeter of Pitch Diameter.
Pinion:
The smaller of two meshed gears.
Pitch Circle:
The circle, the radius of which is equal to the distance from the center of the gear to the pitch point.
Pitch Diameter:
Diameter of the pitch circle
Pitch Point:
The point of tangency of the pitch circles of two meshing gears, where the Line of Centers crosses the pitch circles.
Pressure Angle:
Angle between the Line of Action and a line perpendicular to the Line of Centers.
Profile Shift:
An increase in the Outer Diameter and Root Diameter of a gear, introduced to lower the practical tooth number or acheive a non-standard Center Distance. For a description of Profile Shift, see DANotes.
Ratio:
Ratio of the numbers of teeth on mating gears.
Root Circle:
The circle that passes through the bottom of the tooth spaces.
Root Diameter:
The diameter of the Root Circle.
Stub Gear:
A gear with Stub Teeth.
Stub Tooth:
A gear tooth with only 80% of the usual working depth.
Undercut:
A gear tooth which is thinner at the base than at the Pitch Circle. Caused by too few teeth (too course a Diametral Pitch) for a given Pitch Diameter.
Whole Depth:
The distance between the top of the teeth and the bottom of the spaces between teeth.
Working Depth:
The depth to which a tooth extends into the space between teeth on the mating gear.

## General Notes

Gears that are smaller than 32 teeth for a 14.5° Pressure Angle, or 18 teeth for a 20° Pressure Angle, have a Root Circle smaller than the Base Circle, resulting in the teeth being undercut. Undercutting is avoided by the use of Profile Shift, the shifting the the Addendum and Dedendum outward without altering the location of the base circle. Profile shift is also used to attain non-standard center distances. Gears with shifted profiles will run properly with standard gears.

Standard Pressure Angle is 20°, but 14.5° is traditional. 14.5° was chosen because the sine is almost exactly ¼, which simplified design in the pre-computer age.

There are no "standards" for pitch, but industry has settled on certain common values. Common Diametral Pitches include 48, 32, 24, 20, 18, 16, and 12. Common Modulusises (Modulae? Moduli?) include 0.5, 0.8, 1.00, 1.25, 1.50, 2.50, and 3.

## Formulae

I prefer the term "formulae" to "formulas". I also prefer "octopodes" to "octopi", on the grounds that the suffix "pus" is Greek, and should have a Greek plurization rather than a Latin pluralization, but I digress...

Circular Pitch = π ÷ Diametral Pitch

Diametral Pitch = π ÷ Circular Pitch

Pitch Diameter = Teeth ÷ Diametral Pitch
Pitch Diameter = Teeth × Circular Pitch ÷ pi

Center Distance = (Teeth on Pinion + Teeth on Gear) ÷ (2 × Diametral Pitch)
Center Distance = (Teeth on Pinion + Teeth on Gear) × Circular Pitch ÷ (2 × pi)

Diametral Pitch = (Teeth on Pinion + Teeth on Gear) ÷ (2 × Center Distance)

Circular Pitch = Center Distance × 2 × π ÷ (Teeth on Pinion + Teeth on Gear)

Teeth = Pitch Diameter × Diametral Pitch
Teeth = Pitch Diameter × π ÷ Circular Pitch

Base Circle Diameter = Pitch Diameter × Cosine(Pressure Angle)

### Forumulae Specific to Gears with Standard Teeth

Addendum = 1 ÷ Diametral Pitch
Addendum = 0.3183 × Circular Pitch

Dedendum = 1.157 ÷ Diametral Pitch
Dedendum = 0.3683 × Circular Pitch

Working Depth = 2 ÷ Diametral Pitch
Working Depth = 0.6366 × Circular Pitch

Whole Depth = 2.157 ÷ Diametral Pitch
Whole Depth = 0.6866 × Circular Pitch

Clearance = 0.157 ÷ Diametral Pitch
Clearance = 0.05 × Circular Pitch

Outside Diameter = (Teeth + 2) ÷ Diametral Pitch
Outside Diameter = (Teeth + 2) × Circular Pitch ÷ π

Diametral Pitch = (Teeth + 2) ÷ Outside Diameter

### Forumulae Specific to Gears with Stub Teeth

Addendum = 0.8 ÷ Diametral Pitch
Addendum = 0.2546 × Circular Pitch

Dedendum = 1 ÷ Diametral Pitch
Dedendum = 0.3183 × Circular Pitch

Working Depth = 1.6 ÷ Diametral Pitch
Working Depth = 0.5092 × Circular Pitch

Whole Depth = 1.8 ÷ Diametral Pitch
Whole Depth = 0.5729 × Circular Pitch

Clearance = 0.2 ÷ Diametral Pitch
Clearance = 0.0637 × Circular Pitch

Outside Diameter = (Teeth + 1.6) ÷ Diametral Pitch
Outside Diameter = (Teeth + 1.6) × Circular Pitch ÷ π

Diametral Pitch = (Teeth + 1.6) ÷ Outside Diameter

### Formulae for Metric (Module) Gears

Module = 25.4 ÷ Diametral Pitch (in)

Diametral Pitch (in) = 25.4 ÷ Module

Module = Pitch Diameter ÷ Teeth

Dedendum = 1.157 × Module

Working Depth = 2 × Module

Whole Depth = 2.157 × Module

Pitch Diameter = Module × Teeth

Outside Diameter = Module × (Teeth + 2)

## Identifying Unknown Gears

There are a lot of ways to do this, the obvious one being to use a gear tooth pitch gauge, but I don't have one, so the methods I use are as follows. First:

1. Count the teeth;
3. Divide by the Outer Diameter.

This gives me the Diametral Pitch. It rarely gives me an exact pitch, due to errors in measuring, but it is usually very close to one of the common pitches of 32, 24, 16, etc.

If this gives me a goofy answer like 22.1, I suspect it is a metric gear, and divide 25.4 by the pitch (or divide the pitch by 25.4 and hit the 1/x button on the calculator) and see if it comes reasonably close to one of the standard metric moduli of 0.5, 0.8, 1.0, 1.25, 1.5, 2.5, etc.

If this still gives me a goofy answer like 1.14, then I figure it might be a Stub Gear, and try this:

1. Count the teeth (it doesn't usually change from the last time);
3. Divide by the Outer Diameter (which also doesn't usually change).

It should be close to one of the common pitches, but, alas, it often isn't. If I have a gear with which it meshes, I try this:

1. Add up the teeth on BOTH gears;
2. Divide by two;
3. Divide by the Center Distance.

If this doesn't work, it must be a profile-shifted gear, so I try swearing, which also doesn't work.

## Gear Calculator

 Number of Teeth Diametrical Pitch Pressure Angle Circular Pitch Pitch Diameter Outside Diameter Root Diameter Base Diameter Addendum Dedendum Working Depth Whole Depth