Last updated on Tuesday, July 05, 2022 05:20:05 PM Mountain US Time Zone


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

Gear Theory
Gear Module Data

Spur Gears
Gear Module Formulae
Gearotic Motion


Gear Wheel Designer

Cycloidal gear. Gear Wheel Designer software creates a
variety of clock gear types & generates the 2.5D 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.
This software is 2.5D only. Gearotic Motion is a far
more sophisticated 2.5D & 3D (4th-axis) solution

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.



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1/32" carbide end mill.

 A CNC-made deadbeat escapement wheel that has 30-teeth.

Recoil escapement.


Involute Gears

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.
simple gear can be defined in terms of its pitch,
pressure angle
, & number of teeth.

Gear Definitions

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



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The clear, 3M tape is 0.0025" thick so the final
cut is set to 0.002" below the 0.025" thick, brass plate.


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The gears have no burrs if the last cut ends in the
tape layer. These are module = 1 (25.4 DP) gears.

The fixture returns to the same
position using four 1/4" indexing pins.

A larger, thicker (0.497"), flatter tooling plate was fabricated using MIC6.
The 3/8-16 button-head bolts are below the surface top.
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.

Gearotic Motion

Gearotic Motion (GM) is a CAD/CAM application that is integrated
with Mach3. GM
is a sophisticated 2.5D & 3D (4th-axis)
solution that can
design & mill many gear types including:
involute, lantern, pinion, elliptical, bevel, helical, knuckle, timing, &

 imaginary. It can make short AVI movies of the gear train in motion;
the master (first) gear speed & direction can be controlled.

Periodic reversing motion can be specified over a defined angle.
Different spoke designs plus hand cranks & clock hands are available.

Version program designs the escapement wheel plus the
pendulum with matching pawl for
clock gear trains GM Manual

Spur gear window used to create wheels & pinions using
metric module or imperial diametral pitch specifications.

Lantern wheel & pinion.

GM can design & cut elliptical gears.

A ratchet shape in the ratchets/gadgets window.

A recoil shape in the ratchets/gadgets window.

Graham escapement.

Grasshopper escapement.

Latest Gearotic Motion version (2015).

Escapement wheels made of ABS
plastic & brass using CNC mill.

Gearotic Motion Output Manager.

Involute Gear Cutters

A complete set of eight, M = 0.5 (50.8 DP) involute
gears cutters: high speed tool steel, hardness RC60,

form-relieved cutting teeth, 20 deg pressure angle,
16mm bore with keyway, nominal 40mm OD (Hong Kong).

Modified the supplied packaging
to safely hold the cutter set. The
5/8" wood dowel contacts the lid.

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

Note: metric module & diametral pitch
(DP) cutter numbers are reversed.

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

The cutters are not quite thick enough for secure
clamping so a custom 0.0525" thick washer was made.

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HSS involute cutters: #1, #2, #3, #4, #5, #6, #7, #8.
Pressure Angle (PA) = 20, Module # = 0.5, ID Bore 16mm








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Dividing head with its center & integral dog. The 1-1/2-8
threads are protected by the supplied plastic collar.

Also shown is its adjustable tailstock. Threaded (unfinished)
chuck adapter & additional index plates (top).

dividing_head.jpg (62952 bytes)

gear_cutting.jpg (48963 bytes)
A single HSS toolbit can be ground
to a gear profile. My first try results.
The three-jawed chuck did not hold the stock
perfectly centered.This is easily corrected
using a 4-jaw chuck or collets. I have
now switched to
CNC gears; making them
using the 4th A-axis & involute cutters.










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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 pan> 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
 wrap> 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 deg Pressure Angle (PA) has a better tooth form for cutting pinions than e.g., a 14-1/2 deg 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
Whole Depth
Whole Depth
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.


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 called pedals. Larger gears are termed wheels.

Indicating the rotary table face by performing
adjustments both horizontally & vertically.

Mounting the 4-jaw chuck.

A rotary table alignment bracket is used
to initially center the part before transferring
to the 4th-axis for final tramming.

Gear Theory 
   Gear Module Data
Spur Gears 
Gear Module Formulae 
Gearotic Motion

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