//////////////////////////////////////////////////////////////////////////////////////////////
// LibFile: 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 <BOSL2/std.scad>
// include <BOSL2/involute_gears.scad>
// ```
//////////////////////////////////////////////////////////////////////////////////////////////
// Section: Terminology
// 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.
// Section: Functions
// 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.
// Function: circular_pitch()
// Description: Get tooth density expressed as "circular pitch".
// Arguments:
// mm_per_tooth = Distance between teeth around the pitch circle, in mm.
function circular_pitch(mm_per_tooth=5) = mm_per_tooth;
// Function: diametral_pitch()
// Description: Get tooth density expressed as "diametral pitch".
// Arguments:
// mm_per_tooth = Distance between teeth around the pitch circle, in mm.
function diametral_pitch(mm_per_tooth=5) = PI / mm_per_tooth;
// Function: module_value()
// Description: Get tooth density expressed as "module" or "modulus" in millimeters
// Arguments:
// mm_per_tooth = Distance between teeth around the pitch circle, in mm.
function module_value(mm_per_tooth=5) = mm_per_tooth / PI;
// Function: adendum()
// Description: The height of the gear tooth above the pitch radius.
// Arguments:
// mm_per_tooth = Distance between teeth around the pitch circle, in mm.
function adendum(mm_per_tooth=5) = module_value(mm_per_tooth);
// Function: dedendum()
// Description: The depth of the gear tooth valley, below the pitch radius.
// Arguments:
// mm_per_tooth = Distance between teeth around the pitch circle, in mm.
// clearance = If given, sets the clearance between meshing teeth.
function dedendum(mm_per_tooth=5, clearance=undef) =
(clearance==undef)? (1.25 * module_value(mm_per_tooth)) : (module_value(mm_per_tooth) + clearance);
// Function: pitch_radius()
// Description: Calculates the pitch radius for the gear.
// Arguments:
// mm_per_tooth = Distance between teeth around the pitch circle, in mm.
// number of teeth = The number of teeth on the gear.
function pitch_radius(mm_per_tooth=5, number_of_teeth=11) =
mm_per_tooth * number_of_teeth / PI / 2;
// Function: outer_radius()
// Description:
// Calculates the outer radius for the gear. The gear fits entirely within a cylinder of this radius.
// Arguments:
// 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.
function outer_radius(mm_per_tooth=5, number_of_teeth=11, clearance=undef, interior=false) =
pitch_radius(mm_per_tooth, number_of_teeth) +
(interior? dedendum(mm_per_tooth, clearance) : adendum(mm_per_tooth));
// Function: root_radius()
// Description:
// Calculates the root radius for the gear, at the base of the dedendum.
// Arguments:
// 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.
function root_radius(mm_per_tooth=5, number_of_teeth=11, clearance=undef, interior=false)
= pitch_radius(mm_per_tooth, number_of_teeth) -
(interior? adendum(mm_per_tooth) : dedendum(mm_per_tooth, clearance));
// Function: base_radius()
// Description: Get the base circle for involute teeth.
// Arguments:
// 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.
function base_radius(mm_per_tooth=5, number_of_teeth=11, pressure_angle=28)
= pitch_radius(mm_per_tooth, number_of_teeth) * cos(pressure_angle);
// Section: Modules
// Module: gear_tooth_profile()
// Description:
// Creates the 2D profile for an individual gear tooth.
// Arguments:
// 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(2D):
// gear_tooth_profile(mm_per_tooth=5, number_of_teeth=20, pressure_angle=20);
module gear_tooth_profile(
mm_per_tooth = 3, //this is the "circular pitch", the circumference of the pitch circle divided by the number of teeth
number_of_teeth = 11, //total number of teeth around the entire perimeter
pressure_angle = 28, //Controls how straight or bulged the tooth sides are. In degrees.
backlash = 0.0, //gap between two meshing teeth, in the direction along the circumference of the pitch circle
bevelang = 0.0, //Gear face angle for bevelled gears.
clearance = undef, //gap between top of a tooth on one gear and bottom of valley on a meshing gear (in millimeters)
interior = false
) {
function polar(r,theta) = r*[sin(theta), cos(theta)]; //convert polar to cartesian coordinates
function iang(r1,r2) = sqrt((r2/r1)*(r2/r1) - 1)/PI*180 - acos(r1/r2); //unwind a string this many degrees to go from radius r1 to radius r2
function q7(f,r,b,r2,t,s) = q6(b,s,t,(1-f)*max(b,r)+f*r2); //radius a fraction f up the curved side of the tooth
function q6(b,s,t,d) = polar(d,s*(iang(b,d)+t)); //point at radius d on the involute curve
p = pitch_radius(mm_per_tooth, number_of_teeth);
c = outer_radius(mm_per_tooth, number_of_teeth, clearance, interior);
r = root_radius(mm_per_tooth, number_of_teeth, clearance, interior);
b = base_radius(mm_per_tooth, number_of_teeth, pressure_angle);
t = mm_per_tooth/2-backlash/2; //tooth thickness at pitch circle
k = -iang(b, p) - t/2/p/PI*180; //angle to where involute meets base circle on each side of tooth
scale([1, 1/cos(bevelang), 1])
translate([0,-r,0])
polygon(
points=[
polar(r-1, -181/number_of_teeth),
polar(r, -181/number_of_teeth),
polar(r, r<b ? k : -180/number_of_teeth),
q7(0/5,r,b,c,k, 1),q7(1/5,r,b,c,k, 1),q7(2/5,r,b,c,k, 1),q7(3/5,r,b,c,k, 1),q7(4/5,r,b,c,k, 1),q7(5/5,r,b,c,k, 1),
q7(5/5,r,b,c,k,-1),q7(4/5,r,b,c,k,-1),q7(3/5,r,b,c,k,-1),q7(2/5,r,b,c,k,-1),q7(1/5,r,b,c,k,-1),q7(0/5,r,b,c,k,-1),
polar(r, r<b ? -k : 180/number_of_teeth),
polar(r, 181/number_of_teeth),
polar(r-1, 181/number_of_teeth),
]
);
}
// Module: gear2d()
// 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().
// Arguments:
// 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(2D): Typical Gear Shape
// gear2d(mm_per_tooth=5, number_of_teeth=20);
// Example(2D): Lower Pressure Angle
// gear2d(mm_per_tooth=5, number_of_teeth=20, pressure_angle=20);
// Example(2D): Partial Gear
// gear2d(mm_per_tooth=5, number_of_teeth=20, teeth_to_hide=15, pressure_angle=20);
module gear2d(
mm_per_tooth = 3, //this is the "circular pitch", the circumference of the pitch circle divided by the number of teeth
number_of_teeth = 11, //total number of teeth around the entire perimeter
teeth_to_hide = 0, //number of teeth to delete to make this only a fraction of a circle
pressure_angle = 28, //Controls how straight or bulged the tooth sides are. In degrees.
clearance = undef, //gap between top of a tooth on one gear and bottom of valley on a meshing gear (in millimeters)
backlash = 0.0, //gap between two meshing teeth, in the direction along the circumference of the pitch circle
bevelang = 0.0,
interior = false
) {
r = root_radius(mm_per_tooth, number_of_teeth, clearance, interior);
ang = 360/number_of_teeth/2;
union() {
for (i = [0:number_of_teeth-teeth_to_hide-1] ) {
rotate(i*360/number_of_teeth) {
translate([0,r,0]) {
gear_tooth_profile(
mm_per_tooth = mm_per_tooth,
number_of_teeth = number_of_teeth,
pressure_angle = pressure_angle,
clearance = clearance,
backlash = backlash,
bevelang = bevelang,
interior = interior
);
}
polygon([
[-r*sin(ang), r*cos(ang)],
[0,0],
[r*sin(ang), r*cos(ang)]
]);
}
}
}
}
// Module: gear()
// 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.
// Arguments:
// 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`.
// anchor = Alignment of the gear. Use the constants from `constants.scad`. Default: `CENTER`.
// Example: Spur Gear
// gear(mm_per_tooth=5, number_of_teeth=20, thickness=8, hole_diameter=5);
// Example: 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);
module gear(
mm_per_tooth = 3, //this is the "circular pitch", the circumference of the pitch circle divided by the number of teeth
number_of_teeth = 11, //total number of teeth around the entire perimeter
thickness = 6, //thickness of gear in mm
hole_diameter = 3, //diameter of the hole in the center, in mm
teeth_to_hide = 0, //number of teeth to delete to make this only a fraction of a circle
pressure_angle = 28, //Controls how straight or bulged the tooth sides are. In degrees.
clearance = undef, //gap between top of a tooth on one gear and bottom of valley on a meshing gear (in millimeters)
backlash = 0.0, //gap between two meshing teeth, in the direction along the circumference of the pitch circle
bevelang = 0.0, //angle of bevelled gear face.
twist = undef, //teeth rotate this many degrees from bottom of gear to top. 360 makes the gear a screw with each thread going around once
slices = undef, //Number of slices to divide gear into. Useful for refining gears with `twist`.
interior = false,
orient = ORIENT_Z,
anchor = CENTER
) {
p = pitch_radius(mm_per_tooth, number_of_teeth);
c = outer_radius(mm_per_tooth, number_of_teeth, clearance, interior);
r = root_radius(mm_per_tooth, number_of_teeth, clearance, interior);
p2 = p - (thickness*tan(bevelang));
orient_and_anchor([p, p, thickness], orient, anchor, chain=true) {
difference() {
linear_extrude(height=thickness, center=true, convexity=10, twist=twist, scale=p2/p, slices=slices) {
gear2d(
mm_per_tooth = mm_per_tooth,
number_of_teeth = number_of_teeth,
teeth_to_hide = teeth_to_hide,
pressure_angle = pressure_angle,
clearance = clearance,
backlash = backlash,
bevelang = bevelang,
interior = interior
);
}
if (hole_diameter > 0) {
cylinder(h=2*thickness+1, r=hole_diameter/2, center=true);
}
if (bevelang != 0) {
h = (c-r)*sin(bevelang);
translate([0,0,-thickness/2]) {
difference() {
cube([2*c/cos(bevelang),2*c/cos(bevelang),2*h], center=true);
cylinder(h=h, r1=r, r2=c, center=false);
}
}
}
}
children();
}
}
// Module: rack()
// 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`.
// Arguments:
// 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`.
// anchor = Alignment of the rack. Use the constants from `constants.scad`. Default: `RIGHT`.
// Example:
// rack(mm_per_tooth=5, number_of_teeth=10, thickness=5, height=5, pressure_angle=20);
module rack(
mm_per_tooth = 5, //this is the "circular pitch", the circumference of the pitch circle divided by the number of teeth
number_of_teeth = 20, //total number of teeth along the rack
thickness = 5, //thickness of rack in mm (affects each tooth)
height = 10, //height of rack in mm, from tooth top to back of rack.
pressure_angle = 28, //Controls how straight or bulged the tooth sides are. In degrees.
backlash = 0.0, //gap between two meshing teeth, in the direction along the circumference of the pitch circle
clearance = undef,
orient = ORIENT_X,
anchor = RIGHT
) {
a = adendum(mm_per_tooth);
d = dedendum(mm_per_tooth, clearance);
xa = a * sin(pressure_angle);
xd = d * sin(pressure_angle);
orient_and_anchor([(number_of_teeth-1)*mm_per_tooth, height, thickness], orient, anchor, orig_orient=ORIENT_X, chain=true) {
left((number_of_teeth-1)*mm_per_tooth/2) {
linear_extrude(height = thickness, center = true, convexity = 10) {
for (i = [0:number_of_teeth-1] ) {
translate([i*mm_per_tooth,0,0]) {
polygon(
points=[
[-1/2 * mm_per_tooth - 0.01, a-height],
[-1/2 * mm_per_tooth, -d],
[-1/4 * mm_per_tooth + backlash - xd, -d],
[-1/4 * mm_per_tooth + backlash + xa, a],
[ 1/4 * mm_per_tooth - backlash - xa, a],
[ 1/4 * mm_per_tooth - backlash + xd, -d],
[ 1/2 * mm_per_tooth, -d],
[ 1/2 * mm_per_tooth + 0.01, a-height],
]
);
}
}
}
}
children();
}
}
//////////////////////////////////////////////////////////////////////////////////////////////
//example gear train.
//Try it with OpenSCAD View/Animate command with 20 steps and 24 FPS.
//The gears will continue to be rotated to mesh correctly if you change the number of teeth.
/*
n1 = 11; //red gear number of teeth
n2 = 20; //green gear
n3 = 5; //blue gear
n4 = 20; //orange gear
n5 = 8; //gray rack
mm_per_tooth = 9; //all meshing gears need the same mm_per_tooth (and the same pressure_angle)
thickness = 6;
hole = 3;
height = 12;
d1 =pitch_radius(mm_per_tooth,n1);
d12=pitch_radius(mm_per_tooth,n1) + pitch_radius(mm_per_tooth,n2);
d13=pitch_radius(mm_per_tooth,n1) + pitch_radius(mm_per_tooth,n3);
d14=pitch_radius(mm_per_tooth,n1) + pitch_radius(mm_per_tooth,n4);
translate([ 0, 0, 0]) rotate([0,0, $t*360/n1]) color([1.00,0.75,0.75]) gear(mm_per_tooth,n1,thickness,hole);
translate([ 0, d12, 0]) rotate([0,0,-($t+n2/2-0*n1+1/2)*360/n2]) color([0.75,1.00,0.75]) gear(mm_per_tooth,n2,thickness,hole);
translate([ d13, 0, 0]) rotate([0,0,-($t-n3/4+n1/4+1/2)*360/n3]) color([0.75,0.75,1.00]) gear(mm_per_tooth,n3,thickness,hole);
translate([ d13, 0, 0]) rotate([0,0,-($t-n3/4+n1/4+1/2)*360/n3]) color([0.75,0.75,1.00]) gear(mm_per_tooth,n3,thickness,hole);
translate([-d14, 0, 0]) rotate([0,0,-($t-n4/4-n1/4+1/2-floor(n4/4)-3)*360/n4]) color([1.00,0.75,0.50]) gear(mm_per_tooth,n4,thickness,hole,teeth_to_hide=n4-3);
translate([(-floor(n5/2)-floor(n1/2)+$t+n1/2-1/2)*9, -d1+0.0, 0]) rotate([0,0,0]) color([0.75,0.75,0.75]) rack(mm_per_tooth,n5,thickness,height);
*/
// vim: noexpandtab tabstop=4 shiftwidth=4 softtabstop=4 nowrap