CNC Gears
Last updated on Thursday, March 24, 2011 05:51:48 AM Eastern US Time Zone

Gear Wheel Designer, Deadbeat Escapement, Involute Gears, Mach3, Involute Gear Cutters
Range of Cutters,
Basic Gear Formulae, Module Tooth Dimensions, Common Clock Trains

Gear Wheel Designer

Cycloidal gear. Gear Wheel Designer software creates a variety of clock gear types & generates the g-code.
The units can be metric or imperial, the sequence of operations can be specified, & it will also cut spokes.
It only cuts flat stock so you are limited by the size of the end mill (for small detail) & the depth of the cut.
Used a 1/32" carbide end mill to cut a deadbeat escapement, below. This limited the speed & depth of the cut.

Three curves are used above & below a gear's base circle to design cycloidal gears.

Cycloid. A rack would use true cycloid profiles.

Epicycloid, the shape above the base circle.

Hypocycloid, the shape below the base circle.

Deadbeat Escapement

Deadbeat escapement.

1/32" carbide end mill.

 A CNC deadbeat escapement wheel that has 30-teeth.

Recoil escapement.


Involute Gears

Animated involute of a circle.
The path is depicted by a virtual marker at the end of a taut string as it unwinds off a cylinder (the gear's base circle).

Spur gears incorporate the involute shape. A simple gear can be defined in terms of its pitch, pressure angle, & number of teeth.

Gear Wheel Designer generates G-code for CNC applications, like Mach3. An involute gear example.
The outer dashed circle is the pitch circle & the inner one is the base circle. The involute shape is calculated from the base circle.
The ratio between the two circles is used to derive the pressure angle.


Mach3 CNC control software.

Low-profile fixture (0.1225"). Note the slotted holes in the back for quick, slide-in/slide-out mounting.

After degreasing, the brass plate is mounted using double-sided tape.

The end mill is set to 4.0000" above the brass plate's top surface in the Mach3 Z-axis DRO.

The tape is 0.0025" thick so the final cut is set to 0.002" below the 0.025" thick, brass plate.

The gears have no burrs if the last cut ends in the tape layer. These gears have a module = 1 (25.4T/inch).

The fixture returns to the same position using four ¼" indexing pins.

A larger, thicker (0.497"), flatter tooling plate was fabricated using MIC6. The ⅜-16 hex bolt heads are
below the top surface. Lightly scribed alignment lines mark the plate's center. The outer edges have radii.

Involute gear animation. Not for a clock but another example of Mach3 controlled CNC produced gears.

Involute Gear Cutters

A complete set of eight, M = 0.5 (50.8 DP) involute gears cutters: high speed tool steel, hardness
RC60, precision ground form-relieved cutting teeth, 20° pressure angle, 16mm bore with keyway.
The best overall price in the world for involute gear cutter sets is CTC Tools in Hong Kong.


Range of Cutters
To cut gears from use cutter number
12 to 13 teeth 8
14 to 16 teeth 7
17 to 20 teeth 6
21 to 25 teeth 5
26 to 34 teeth 4
35 to 54 teeth 3
55 to 134 teeth 2
135 teeth to a rack 1

Proxxon 24 425 black-oxide HSS 16mm arbor for involute gear cutters (Germany).
This arbor was designed for small-module cutters (e.g., M = 0.5, 0.6, 0.7, 0.8).

Proxxon 24 425 Arbor


Dimensions (mm)
Arbor Size 16 x 5
Key Size 4 (W) x 5 (L)
Wrench Size 17
Thread M8-1.25 x 12
Shank Diameter 10
Shank Length 30
Body Diameter 24.7
Body Length 25
Overall Length 65


Basic Gear Formulae
Gear (G) Dimensions Examples
Imperial Diametral Pitch (DP) = # teeth/inch 24 teeth/inch
Outside Diameter (OD) inches = (G1+2)/DP (40+2)/24 = 1.75 inches outside diameter
Whole Tooth Depth inches = 2.157/DP 2.157/24 = 0.089875 inches depth
Distance Between Gear Centers = (G1+G2)/(2*DP) (30+20)/(2*24) = 1.0417" distance
Metric Metric Module (M) = 25.4/DP 25.4/28.222 = 0.9
DP = 25.4/M 25.4/0.9 = 28.222 teeth per inch
Circular Pitch (CP) mm = M*pi 1*3.142 = 3.142 mm
M = (CP)/pi 2.827 mm / 3.142 = 0.9
Outside Diameter (OD) mm = (G1+2)*M (24+2)*0.9 = 23.4 mm outside diameter
Whole Tooth Depth mm = 2.157*M 2.157*0.9 = 1.9413 mm depth

Formulae Table Notes
 (a) Diametral Pitch (DP) is measured at the Pitch Circle (PC), not the Outside Diameter (OD). G1 = # of teeth for Gear 1.
The metric standard measure for pitch is the Module (M). (c) The distance between centers is determined by the
total number of teeth used for the mating pair of gears. Therefore, any G1+G2 combination that that sums to the same
value (in the numerator) yields the same distance.
(d) The British standardized the calculations by starting with a
diameter of 1" making the circumference equal to pi (3.1416). This simplifies the calculations as pi is a constant on
both sides of the equation for calculating blank diameters & thus, cancel. Adding 2 to the teeth number works
because gears contact each other at exactly one-half the distance from the top of the gear to the bottom. There is
also a bottom clearance but it is not considered in simple calculations. (e) 2.157 is a constant. (f) 1 mm = 0.03937" &
1" = 25.4 mm. (g) A 20° Pressure Angle (PA) has a better tooth form for cutting pinions than e.g., a 14½° PA.

To run two gears together, their pitches & pressure angles must match. (h)
The contact point/plane on a true
 involute generated on & from the Pitch Circle is between the addenda, only.

Module Pitch Tooth Dimensions
M DP CP (mm) CP ( inches) Addendum (mm) Dedendum † (mm) Whole Depth † (mm) Whole Depth ‡ (mm)
0.5 50.800 1.571 0.0618 0.50 0.583 1.083 1.079
0.6 42.333 1.885 0.0742 0.60 0.700 1.300 1.294
0.7 36.286 2.199 0.0865 0.70 0.817 1.517 1.510
0.8 31.750 2.513 0.0989 0.80 0.933 1.733 1.726
0.9 28.222 2.827 0.1113 0.90 1.050 1.950 1.941
1.0 25.400 pi 0.1237 1 1.167 2.167 2.157

Dedendum & total depth when clearance = 0.1666 x module or one-sixth module.
Total depth equivalent to American standard full-depth teeth where clearance = 0.157 x module.

Module (M) - The length, in mm, of the pitch circle diameter per tooth.
Diametral Pitch (DP) - The number of teeth per one inch of pitch circle diameter.
Circular Pitch (CP) - The distance between adjacent teeth measured along the are at the pitch circle diameter
Addendum - The height of the tooth above the pitch circle diameter.
Center Distance - The distance between the axes of two gears in mesh.
Circular Tooth Thickness - The width of a tooth measured along the are at the pitch circle diameter.
Dedendum - The depth of the tooth below the pitch circle diameter.
Outside Diameter (OD) - The outside diameter of the gear.
Base Circle Diameter  - The diameter on which the involute teeth profile is based.
Pitch Circle Diameter (PCD) - The diameter of the pitch circle.
Pitch Point - The point at which the pitch circle diameters of two gears in mesh coincide.
Pitch to Back - The distance on a rack between the pitch circle diameter line & the rear face of the rack.
Pressure Angle (PA) - The angle between the tooth profile at the pitch circle diameter & a radial line passing through the same point.
Whole Depth - The total depth of the space between adjacent teeth.

Common Clock Trains
Center Wheel 3rd Pinion 3rd Wheel Escapement Pinion Escapement Wheel Vibrations per Minute Pendulum Length (inches)
112 14 105 14 60 60 39.14
96 12 90 12 30 60 39.14
80 10 75 10 30 60 39.14
64 8 60 8 30 60 39.14
75 8 60 8 32 75 25.53
80 8 72 8 30 90 17.39
108 12 100 10 32 96 15.28

Drawing M = 0.5 (50.8 DP) 12T, 90T, 96T involute, a 30T ratchet, & a 30T deadbeat clock gears using a
pen on cardboard to visualize sizes. In clock gear nomenclature small gears are called pinions & their
teeth are called pedals while the larger gears are termed wheels.

Gear Theory    Gear Module Data    Spur Gears    Gear Module Formulae    See "Gears & Gear Cutting" by Ivan Law

Gear Wheel Designer, Deadbeat Escapement, Involute Gears, Mach3, Involute Gear Cutters
Range of Cutters,
Basic Gear Formulae, Module Tooth Dimensions, Common Clock Trains