involute_gears.scad
Involute Spur Gears and Racks
by Leemon Baird, 2011, Leemon@Leemon.com http://www.thingiverse.com/thing:5505
Additional fixes and improvements by Revar Desmera, 2017-2019, revarbat@gmail.com
This file is public domain. Use it for any purpose, including commercial applications. Attribution would be nice, but is not required. There is no warranty of any kind, including its correctness, usefulness, or safety.
This is parameterized involute spur (or helical) gear. It is much simpler and less powerful than others on Thingiverse. But it is public domain. I implemented it from scratch from the descriptions and equations on Wikipedia and the web, using Mathematica for calculations and testing, and I now release it into the public domain.
To use, add the following line to the beginning of your file:
include <BOSL/constants.scad>
use <BOSL/involute_gears.scad>
The outline of a gear is a smooth circle (the "pitch circle") which has mountains and valleys added so it is toothed. There is an inner circle (the "root circle") that touches the base of all the teeth, an outer circle that touches the tips of all the teeth, and the invisible pitch circle in between them. There is also a "base circle", which can be smaller than all three of the others, which controls the shape of the teeth. The side of each tooth lies on the path that the end of a string would follow if it were wrapped tightly around the base circle, then slowly unwound. That shape is an "involute", which gives this type of gear its name.
These functions let the user find the derived dimensions of the gear. A gear fits within a circle of radius outer_radius, and two gears should have their centers separated by the sum of their pitch_radius.
Description: Get tooth density expressed as "circular pitch".
| Argument | What it does |
|---|---|
mm_per_tooth |
Distance between teeth around the pitch circle, in mm. |
Description: Get tooth density expressed as "diametral pitch".
| Argument | What it does |
|---|---|
mm_per_tooth |
Distance between teeth around the pitch circle, in mm. |
Description: Get tooth density expressed as "module" or "modulus" in millimeters
| Argument | What it does |
|---|---|
mm_per_tooth |
Distance between teeth around the pitch circle, in mm. |
Description: The height of the gear tooth above the pitch radius.
| Argument | What it does |
|---|---|
mm_per_tooth |
Distance between teeth around the pitch circle, in mm. |
Description: The depth of the gear tooth valley, below the pitch radius.
| Argument | What it does |
|---|---|
mm_per_tooth |
Distance between teeth around the pitch circle, in mm. |
clearance |
If given, sets the clearance between meshing teeth. |
Description: Calculates the pitch radius for the gear.
| Argument | What it does |
|---|---|
mm_per_tooth |
Distance between teeth around the pitch circle, in mm. |
number of teeth |
The number of teeth on the gear. |
Description: Calculates the outer radius for the gear. The gear fits entirely within a cylinder of this radius.
| Argument | What it does |
|---|---|
mm_per_tooth |
Distance between teeth around the pitch circle, in mm. |
number of teeth |
The number of teeth on the gear. |
clearance |
If given, sets the clearance between meshing teeth. |
interior |
If true, calculate for an interior gear. |
Description: Calculates the root radius for the gear, at the base of the dedendum.
| Argument | What it does |
|---|---|
mm_per_tooth |
Distance between teeth around the pitch circle, in mm. |
number of teeth |
The number of teeth on the gear. |
clearance |
If given, sets the clearance between meshing teeth. |
interior |
If true, calculate for an interior gear. |
Description: Get the base circle for involute teeth.
| Argument | What it does |
|---|---|
mm_per_tooth |
Distance between teeth around the pitch circle, in mm. |
number_of_teeth |
The number of teeth on the gear. |
pressure_angle |
Pressure angle in degrees. Controls how straight or bulged the tooth sides are. |
Description: Creates the 2D profile for an individual gear tooth.
| Argument | What it does |
|---|---|
mm_per_tooth |
This is the "circular pitch", the circumference of the pitch circle divided by the number of teeth |
number_of_teeth |
Total number of teeth along the rack |
pressure_angle |
Controls how straight or bulged the tooth sides are. In degrees. |
backlash |
Gap between two meshing teeth, in the direction along the circumference of the pitch circle |
bevelang |
Angle of beveled gear face. |
clearance |
Gap between top of a tooth on one gear and bottom of valley on a meshing gear (in millimeters) |
interior |
If true, create a mask for difference()ing from something else. |
Example:
gear_tooth_profile(mm_per_tooth=5, number_of_teeth=20, pressure_angle=20);
Description: Creates a 2D involute spur gear, with reasonable defaults for all the parameters. Normally, you should just specify the first 2 parameters, and let the rest be default values. Meshing gears must match in mm_per_tooth, pressure_angle, and twist, and be separated by the sum of their pitch radii, which can be found with pitch_radius().
| Argument | What it does |
|---|---|
mm_per_tooth |
This is the "circular pitch", the circumference of the pitch circle divided by the number of teeth |
number_of_teeth |
Total number of teeth along the rack |
teeth_to_hide |
Number of teeth to delete to make this only a fraction of a circle |
pressure_angle |
Controls how straight or bulged the tooth sides are. In degrees. |
clearance |
Gap between top of a tooth on one gear and bottom of valley on a meshing gear (in millimeters) |
backlash |
Gap between two meshing teeth, in the direction along the circumference of the pitch circle |
bevelang |
Angle of beveled gear face. |
interior |
If true, create a mask for difference()ing from something else. |
Example 1: Typical Gear Shape
gear2d(mm_per_tooth=5, number_of_teeth=20);
Example 2: Lower Pressure Angle
gear2d(mm_per_tooth=5, number_of_teeth=20, pressure_angle=20);
Example 3: Partial Gear
gear2d(mm_per_tooth=5, number_of_teeth=20, teeth_to_hide=15, pressure_angle=20);
Description:
Creates a (potentially helical) involute spur gear.
The module gear() gives an involute spur gear, with reasonable
defaults for all the parameters. Normally, you should just choose
the first 4 parameters, and let the rest be default values. The
module gear() gives a gear in the XY plane, centered on the origin,
with one tooth centered on the positive Y axis. The various functions
below it take the same parameters, and return various measurements
for the gear. The most important is pitch_radius(), which tells
how far apart to space gears that are meshing, and outer_radius(),
which gives the size of the region filled by the gear. A gear has
a "pitch circle", which is an invisible circle that cuts through
the middle of each tooth (though not the exact center). In order
for two gears to mesh, their pitch circles should just touch. So
the distance between their centers should be pitch_radius() for
one, plus pitch_radius() for the other, which gives the radii of
their pitch circles.
In order for two gears to mesh, they must have the same mm_per_tooth
and pressure_angle parameters. mm_per_tooth gives the number
of millimeters of arc around the pitch circle covered by one tooth
and one space between teeth. The pressure_angle controls how flat or
bulged the sides of the teeth are. Common values include 14.5
degrees and 20 degrees, and occasionally 25. Though I've seen 28
recommended for plastic gears. Larger numbers bulge out more, giving
stronger teeth, so 28 degrees is the default here.
The ratio of number_of_teeth for two meshing gears gives how many
times one will make a full revolution when the the other makes one
full revolution. If the two numbers are coprime (i.e. are not
both divisible by the same number greater than 1), then every tooth
on one gear will meet every tooth on the other, for more even wear.
So coprime numbers of teeth are good.
| Argument | What it does |
|---|---|
mm_per_tooth |
This is the "circular pitch", the circumference of the pitch circle divided by the number of teeth |
number_of_teeth |
Total number of teeth around the entire perimeter |
thickness |
Thickness of gear in mm |
hole_diameter |
Diameter of the hole in the center, in mm |
teeth_to_hide |
Number of teeth to delete to make this only a fraction of a circle |
pressure_angle |
Controls how straight or bulged the tooth sides are. In degrees. |
clearance |
Clearance gap at the bottom of the inter-tooth valleys. |
backlash |
Gap between two meshing teeth, in the direction along the circumference of the pitch circle |
bevelang |
Angle of beveled gear face. |
twist |
Teeth rotate this many degrees from bottom of gear to top. 360 makes the gear a screw with each thread going around once. |
slices |
Number of vertical layers to divide gear into. Useful for refining gears with twist. |
scale |
Scale of top of gear compared to bottom. Useful for making crown gears. |
interior |
If true, create a mask for difference()ing from something else. |
orient |
Orientation of the gear. Use the ORIENT_ constants from constants.scad. Default: ORIENT_Z. |
align |
Alignment of the gear. Use the V_ constants from constants.scad. Default: V_CENTER. |
Example 1: Spur Gear
gear(mm_per_tooth=5, number_of_teeth=20, thickness=8, hole_diameter=5);
Example 2: Beveled Gear
gear(mm_per_tooth=5, number_of_teeth=20, thickness=10*cos(45), hole_diameter=5, twist=-30, bevelang=45, slices=12, $fa=1, $fs=1);
Description:
The module rack() gives a rack, which is a bar with teeth. A
rack can mesh with any gear that has the same mm_per_tooth and
pressure_angle.
| Argument | What it does |
|---|---|
mm_per_tooth |
This is the "circular pitch", the circumference of the pitch circle divided by the number of teeth |
number_of_teeth |
Total number of teeth along the rack |
thickness |
Thickness of rack in mm (affects each tooth) |
height |
Height of rack in mm, from tooth top to back of rack. |
pressure_angle |
Controls how straight or bulged the tooth sides are. In degrees. |
backlash |
Gap between two meshing teeth, in the direction along the circumference of the pitch circle |
orient |
Orientation of the rack. Use the ORIENT_ constants from constants.scad. Default: ORIENT_X. |
align |
Alignment of the rack. Use the V_ constants from constants.scad. Default: V_RIGHT. |
Example:
rack(mm_per_tooth=5, number_of_teeth=10, thickness=5, height=5, pressure_angle=20);
