Gears
Last updated on
Monday, February 15, 2021 06:49:03 AM
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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 gcode.
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 (4thaxis) 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.
Click on linked thumbnails
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1/32" carbide end mill.
A CNCmade deadbeat escapement wheel
that has 30teeth.
Recoil escapement.
Ratchet.
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.
A
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 oneinch 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 Gcode
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
Mach3 CNC software.
Lowprofile fixture (0.1225"). Note the slotted
holes
in the back for quick, slidein/slideout mounting.
After degreasing, the brass plate is
mounted using
doublesided tape.
The end mill is set to 4.0000" above the brass
plate's top
surface in the Mach3 Zaxis 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 3/816 buttonhead 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 (4thaxis)
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 4.xxx 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,
formrelieved 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 numberuse DP
cutter number12 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 blackoxide HSS 16mm
arbor for involute gear cutters
(Germany).
This arbor was designed for smallmodule
cutters (e.g., M = 0.5, 0.6, 0.7,
0.8).
Proxxon 24 425 Arbor
Attributes
Dimensions (mm) Arbor Size 16 x 5 Key Size 4 (W) x 5 (L) Wrench Size 17 Thread M81.25 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.
Click on linked thumbnails
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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
Click on linked thumbnails
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Dividing head with its center & integral dog. The 11/28
threads are protected by the supplied plastic collar.
Also shown is its adjustable tailstock. Threaded (unfinished)
chuck adapter
& additional index plates (top).
A single HSS toolbit can be ground
to a gear profile. My first try results.
The threejawed chuck did not hold the stock
perfectly centered.This is easily corrected
using a 4jaw chuck or collets. I have
now switched to
CNC gears; making
them
using the 4th Aaxis & involute cutters.
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 = 1.75inches outside diameter  
Whole Tooth Depth inches = 2.157/DP pan>  2.157/24 = 0.089875inches 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.
(b)
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 onehalf 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 141/2 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 M DP CP
(mm)CP
(inches)Addendum
(mm)Dedendum
(mm)Whole Depth
(mm)Whole Depth
(mm)0.5 50.800 1.5710.0618 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 onesixth module.
Total depth equivalent to
American standard fulldepth 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 clockgear
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 4jaw chuck.
A
rotary table
alignment bracket
is used
to
initially center the
part before
transferring
to the
4thaxis 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