Gears
Last updated on
Thursday, March 17, 2022 06:25:00 AM
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.
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 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
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 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,
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 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 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
Attributes
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).
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.
(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 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.
