Gear Theory
Here are some links and notes I made when researching how to make gears on a CNC machine.
If you just want to make a gear without having to read and understand all the theory below, then just go to the program section Clock & Gear Making References of this page for some really cool software tools that will get you up and running with making gears in no time. There are many shareware programs and Java calculators that work really well. These details and links on the page are provided just for my convenience and a reference, but feel free to use them too - at your own risk.
Terminology
Addendum ( h a ) = The height of the tooth above the pitch circle diameter.
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 diameter ( D b ) = The diameter on which the involute teeth profile is based
Centre distance (a) = The distance between the axes of two gears in mesh.
Chordal Addendum = 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 pitch (p) = The distance between adjacent teeth measured along the are at the pitch circle diameter
Circular tooth thickness (ctt) = The width of a tooth measured along the are at the pitch circle diameter.
Circular Thickness = The thickness of the tooth measured along an arc following the Pitch
CircleClearance = 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 ( h f ) = The depth of the tooth below the pitch circle diameter.
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.
Diametral pitch (d p ) = The number of teeth per one inch of pitch circle diameter.
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
Module. (m) = The length, in mm, of the pitch circle diameter per tooth
Outside diameter ( D o ) = The outside diameter of the gear
Pitch to back = The distance on a rack between the pitch circle diameter line and the rear face of the rack
Pitch Circle = The circle, the radius of which is equal to the distance from the center of the gear to the pitch point
Pitch circle dia ( p ) = The diameter of the pitch circle
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
Pitch point. = The point at which the pitch circle diameters of two gears in mesh coincide
Pressure Angle = Angle between the Line of Action and a line perpendicular to the Line of Centres
Pressure angle = The angle between the tooth profile at the pitch circle diameter and a radial line passing through the same point
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
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
Whole depth = The total depth of the space between adjacent teeth.
Working Depth = The depth to which a tooth extends into the space between teeth on the mating gear
Some Calculations:
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
Addendum = Module
Dedendum = 1.157 × Module
Working Depth = 2 × Module
Whole Depth = 2.157 × Module
Pitch Diameter = Module × Teeth
Outside Diameter = Module × (Teeth + 2)
