Gear Theory

• Clock & Gear Making References

• Colin's Clock making background

• Colin's Wooden Clocks

• Colin's Clocks Downloads

• Making Gears using CNC Machines

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)