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publicDomainGearV1.1.scad
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//////////////////////////////////////////////////////////////////////////////////////////////
// Public Domain Parametric Involute Spur Gear (and involute helical gear and involute rack)
// version 1.1
// by Leemon Baird, 2011, Leemon@Leemon.com
//http://www.thingiverse.com/thing:5505
//
// 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.
//
// http://en.wikipedia.org/wiki/Involute_gear
// http://en.wikipedia.org/wiki/Gear
// http://en.wikipedia.org/wiki/List_of_gear_nomenclature
// http://gtrebaol.free.fr/doc/catia/spur_gear.html
// http://www.cs.cmu.edu/~rapidproto/mechanisms/chpt7.html
//
// 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 adendum_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 pitch 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.
//
// 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.
//
// Some terminology:
// The outline of a gear is a smooth circle (the "pitch circle") which has mountains and valleys
// added so it is toothed. So 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.
//
//////////////////////////////////////////////////////////////////////////////////////////////
//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.
//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().
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
twist = 0, //teeth rotate this many degrees from bottom of gear to top. 360 makes the gear a screw with each thread going around once
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 = 0.0, //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
) {
pi = 3.1415926;
p = mm_per_tooth * number_of_teeth / pi / 2; //radius of pitch circle
c = p + mm_per_tooth / pi - clearance; //radius of outer circle
b = p*cos(pressure_angle); //radius of base circle
r = p-(c-p)-clearance; //radius of root circle
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
difference() {
for (i = [0:number_of_teeth-teeth_to_hide-1] )
rotate([0,0,i*360/number_of_teeth])
linear_extrude(height = thickness, center = true, convexity = 10, twist = twist)
polygon(
points=[
[0, -hole_diameter/10],
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)
],
paths=[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]]
);
cylinder(h=2*thickness+1, r=hole_diameter/2, center=true, $fn=20);
}
};
//these 4 functions are used by gear
function polar(r,theta) = r*[sin(theta), cos(theta)]; //convert polar to cartesian coordinates
function iang(r1,r2) = sqrt((r2/r1)*(r2/r1) - 1)/3.1415926*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
//a rack, which is a straight line with teeth (the same as a segment from a giant gear with a huge number of teeth).
//The "pitch circle" is a line along the X axis.
module rack (
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 along the rack
thickness = 6, //thickness of rack in mm (affects each tooth)
height = 120, //height of rack in mm, from tooth top to far side 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
) {
pi = 3.1415926;
a = mm_per_tooth / pi; //addendum
t = a*cos(pressure_angle)-1; //tooth side is tilted so top/bottom corners move this amount
for (i = [0:number_of_teeth-1] )
translate([i*mm_per_tooth,0,0])
linear_extrude(height = thickness, center = true, convexity = 10)
polygon(
points=[
[-mm_per_tooth * 3/4, a-height],
[-mm_per_tooth * 3/4 - backlash, -a],
[-mm_per_tooth * 1/4 + backlash - t, -a],
[-mm_per_tooth * 1/4 + backlash + t, a],
[ mm_per_tooth * 1/4 - backlash - t, a],
[ mm_per_tooth * 1/4 - backlash + t, -a],
[ mm_per_tooth * 3/4 + backlash, -a],
[ mm_per_tooth * 3/4, a-height],
],
paths=[[0,1,2,3,4,5,6,7]]
);
};
//These 5 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 pictch_radius.
function circular_pitch (mm_per_tooth=3) = mm_per_tooth; //tooth density expressed as "circular pitch" in millimeters
function diametral_pitch (mm_per_tooth=3) = 3.1415926 / mm_per_tooth; //tooth density expressed as "diametral pitch" in teeth per millimeter
function module_value (mm_per_tooth=3) = mm_per_tooth / pi; //tooth density expressed as "module" or "modulus" in millimeters
function pitch_radius (mm_per_tooth=3,number_of_teeth=11) = mm_per_tooth * number_of_teeth / 3.1415926 / 2;
function outer_radius (mm_per_tooth=3,number_of_teeth=11,clearance=0.1) //The gear fits entirely within a cylinder of this radius.
= mm_per_tooth*(1+number_of_teeth/2)/3.1415926 - clearance;
//////////////////////////////////////////////////////////////////////////////////////////////
//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,0,108);
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,-($t-n4/4-n1/4+1/2-floor(n4/4)-3)*360/n4]) color([1.00,0.75,0.50]) gear(mm_per_tooth,n4,thickness,hole,0,n4-3);
translate([(-floor(n5/2)-floor(n1/2)+$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