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.
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)
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
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
Module = 25.4 ÷ Diametral Pitch (in)
Diametral Pitch (in) = 25.4 ÷ Module
Module = Pitch Diameter ÷ Teeth
Addendum = Module
Dedendum = 1.157 × Module
Working Depth = 2 × Module
Whole Depth = 2.157 × Module
Pitch Diameter = Module × Teeth
Outside Diameter = Module × (Teeth + 2)
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:
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:
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:
If this doesn't work, it must be a profile-shifted gear, so I try swearing, which also doesn't work.
© 2005 W. E. Johns