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//////////////////////////////////////////////////////////////////////////////////////////////
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// Public Domain Parametric Involute Spur Gear (and involute helical gear and involute rack)
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// version 1.1
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// by Leemon Baird, 2011, Leemon@Leemon.com
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//http://www.thingiverse.com/thing:5505
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//
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// This file is public domain. Use it for any purpose, including commercial
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// applications. Attribution would be nice, but is not required. There is
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// no warranty of any kind, including its correctness, usefulness, or safety.
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//
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// This is parameterized involute spur (or helical) gear. It is much simpler and less powerful than
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// others on Thingiverse. But it is public domain. I implemented it from scratch from the
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// descriptions and equations on Wikipedia and the web, using Mathematica for calculations and testing,
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// and I now release it into the public domain.
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//
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// http://en.wikipedia.org/wiki/Involute_gear
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// http://en.wikipedia.org/wiki/Gear
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// http://en.wikipedia.org/wiki/List_of_gear_nomenclature
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// http://gtrebaol.free.fr/doc/catia/spur_gear.html
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// http://www.cs.cmu.edu/~rapidproto/mechanisms/chpt7.html
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//
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// The module gear() gives an involute spur gear, with reasonable defaults for all the parameters.
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// Normally, you should just choose the first 4 parameters, and let the rest be default values.
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// The module gear() gives a gear in the XY plane, centered on the origin, with one tooth centered on
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// the positive Y axis. The various functions below it take the same parameters, and return various
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// measurements for the gear. The most important is pitch_radius, which tells how far apart to space
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// gears that are meshing, and adendum_radius, which gives the size of the region filled by the gear.
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// A gear has a "pitch circle", which is an invisible circle that cuts through the middle of each
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// tooth (though not the exact center). In order for two gears to mesh, their pitch circles should
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// just touch. So the distance between their centers should be pitch_radius() for one, plus pitch_radius()
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// for the other, which gives the radii of their pitch circles.
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//
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// In order for two gears to mesh, they must have the same mm_per_tooth and pressure_angle parameters.
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// mm_per_tooth gives the number of millimeters of arc around the pitch circle covered by one tooth and one
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// space between teeth. The pitch angle controls how flat or bulged the sides of the teeth are. Common
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// values include 14.5 degrees and 20 degrees, and occasionally 25. Though I've seen 28 recommended for
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// plastic gears. Larger numbers bulge out more, giving stronger teeth, so 28 degrees is the default here.
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//
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// The ratio of number_of_teeth for two meshing gears gives how many times one will make a full
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// revolution when the the other makes one full revolution. If the two numbers are coprime (i.e.
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// are not both divisible by the same number greater than 1), then every tooth on one gear
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// will meet every tooth on the other, for more even wear. So coprime numbers of teeth are good.
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//
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// The module rack() gives a rack, which is a bar with teeth. A rack can mesh with any
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// gear that has the same mm_per_tooth and pressure_angle.
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//
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// Some terminology:
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// The outline of a gear is a smooth circle (the "pitch circle") which has mountains and valleys
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// added so it is toothed. So there is an inner circle (the "root circle") that touches the
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// base of all the teeth, an outer circle that touches the tips of all the teeth,
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// and the invisible pitch circle in between them. There is also a "base circle", which can be smaller than
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// all three of the others, which controls the shape of the teeth. The side of each tooth lies on the path
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// that the end of a string would follow if it were wrapped tightly around the base circle, then slowly unwound.
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// That shape is an "involute", which gives this type of gear its name.
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//
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//////////////////////////////////////////////////////////////////////////////////////////////
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//An involute spur gear, with reasonable defaults for all the parameters.
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//Normally, you should just choose the first 4 parameters, and let the rest be default values.
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//Meshing gears must match in mm_per_tooth, pressure_angle, and twist,
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//and be separated by the sum of their pitch radii, which can be found with pitch_radius().
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module gear (
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mm_per_tooth = 3, //this is the "circular pitch", the circumference of the pitch circle divided by the number of teeth
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number_of_teeth = 11, //total number of teeth around the entire perimeter
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thickness = 6, //thickness of gear in mm
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hole_diameter = 3, //diameter of the hole in the center, in mm
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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
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teeth_to_hide = 0, //number of teeth to delete to make this only a fraction of a circle
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pressure_angle = 28, //Controls how straight or bulged the tooth sides are. In degrees.
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clearance = 0.0, //gap between top of a tooth on one gear and bottom of valley on a meshing gear (in millimeters)
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backlash = 0.0, //gap between two meshing teeth, in the direction along the circumference of the pitch circle
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cut_circles = 0
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) {
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assign(pi = 3.1415926)
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assign(p = mm_per_tooth * number_of_teeth / pi / 2) //radius of pitch circle
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assign(c = p + mm_per_tooth / pi - clearance) //radius of outer circle
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assign(b = p*cos(pressure_angle)) //radius of base circle
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assign(r = p-(c-p)-clearance) //radius of root circle
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assign(t = mm_per_tooth/2-backlash/2) //tooth thickness at pitch circle
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assign(k = -iang(b, p) - t/2/p/pi*180) { //angle to where involute meets base circle on each side of tooth
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difference() {
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union() {
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for (i = [0:number_of_teeth-teeth_to_hide-1] )
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rotate([0,0,i*360/number_of_teeth])
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linear_extrude(height = thickness, center = true, convexity = 10, twist = twist)
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polygon(
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points=[
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[0, -hole_diameter/10],
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polar(r, -181/number_of_teeth),
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polar(r, r<b ? k : -180/number_of_teeth),
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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),
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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),
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polar(r, r<b ? -k : 180/number_of_teeth),
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polar(r, 181/number_of_teeth)
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],
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paths=[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]]
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);
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cylinder(r=b,h=thickness, center=true);
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}
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cylinder(h=2*thickness+1, r=hole_diameter/2, center=true, $fn=20);
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}
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}
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};
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//these 4 functions are used by gear
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function polar(r,theta) = r*[sin(theta), cos(theta)]; //convert polar to cartesian coordinates
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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
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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
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function q6(b,s,t,d) = polar(d,s*(iang(b,d)+t)); //point at radius d on the involute curve
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//a rack, which is a straight line with teeth (the same as a segment from a giant gear with a huge number of teeth).
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//The "pitch circle" is a line along the X axis.
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module rack (
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mm_per_tooth = 3, //this is the "circular pitch", the circumference of the pitch circle divided by the number of teeth
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number_of_teeth = 11, //total number of teeth along the rack
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thickness = 6, //thickness of rack in mm (affects each tooth)
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height = 120, //height of rack in mm, from tooth top to far side of rack.
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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
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pressure_angle = 28, //Controls how straight or bulged the tooth sides are. In degrees.
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backlash = 0.0 //gap between two meshing teeth, in the direction along the circumference of the pitch circle
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) {
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pi = 3.1415926;
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a = mm_per_tooth / pi; //addendum
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t = a*tan(pressure_angle); //tooth side is tilted so top/bottom corners move this amount
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shift = -thickness*2*tan(twist);
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union() {
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for (i = [0:number_of_teeth-1] )
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translate([i*mm_per_tooth,0,-thickness/2])
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//linear_extrude(height = thickness, center = true, convexity = 10)
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polyhedron(
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points=[
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[-mm_per_tooth * 3/4, a-height, 0], // 0
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[-mm_per_tooth * 3/4 - backlash, -a, 0],
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[-mm_per_tooth * 1/4 + backlash - t, -a, 0],
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[-mm_per_tooth * 1/4 + backlash + t, a, 0],
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[ mm_per_tooth * 1/4 - backlash - t, a, 0],
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[ mm_per_tooth * 1/4 - backlash + t, -a, 0],
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[ mm_per_tooth * 3/4 + backlash, -a, 0],
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[ mm_per_tooth * 3/4, a-height, 0], // 7
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[-mm_per_tooth * 3/4 + shift, a-height, thickness],
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[-mm_per_tooth * 3/4 - backlash + shift, -a, thickness],
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[-mm_per_tooth * 1/4 + backlash - t + shift, -a, thickness],
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[-mm_per_tooth * 1/4 + backlash + t + shift, a, thickness],
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[ mm_per_tooth * 1/4 - backlash - t + shift, a, thickness],
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[ mm_per_tooth * 1/4 - backlash + t + shift, -a, thickness],
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[ mm_per_tooth * 3/4 + backlash + shift, -a, thickness],
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[ mm_per_tooth * 3/4 + shift, a-height, thickness],
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],
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faces=[
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[7,6,5,4,3,2,1,0], // bottom
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[0,1,9,8], // side
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[1,2,10,9],
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[2,3,11,10],
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[3,4,12,11],
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[4,5,13,12],
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[5,6,14,13],
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[6,7,15,14], // side
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[7,0,8,15], // back
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[8,9,10,11,12,13,14,15], // top
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]);
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translate([(number_of_teeth-2)*mm_per_tooth/2,-(height)/2,0]) cube([(number_of_teeth+5)*mm_per_tooth,height-2*a,thickness], center=true);
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}
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};
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//These 5 functions let the user find the derived dimensions of the gear.
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//A gear fits within a circle of radius outer_radius, and two gears should have
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//their centers separated by the sum of their pictch_radius.
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function circular_pitch (mm_per_tooth=3) = mm_per_tooth; //tooth density expressed as "circular pitch" in millimeters
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function diametral_pitch (mm_per_tooth=3) = 3.1415926 / mm_per_tooth; //tooth density expressed as "diametral pitch" in teeth per millimeter
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function module_value (mm_per_tooth=3) = mm_per_tooth / pi; //tooth density expressed as "module" or "modulus" in millimeters
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function pitch_radius (mm_per_tooth=3,number_of_teeth=11) = mm_per_tooth * number_of_teeth / 3.1415926 / 2;
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function outer_radius (mm_per_tooth=3,number_of_teeth=11,clearance=0.1) //The gear fits entirely within a cylinder of this radius.
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= mm_per_tooth*(1+number_of_teeth/2)/3.1415926 - clearance;
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//////////////////////////////////////////////////////////////////////////////////////////////
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//example gear train.
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//Try it with OpenSCAD View/Animate command with 20 steps and 24 FPS.
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//The gears will continue to be rotated to mesh correctly if you change the number of teeth.
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inner_screw=1;
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module servo() {
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$fn=50;
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translate([6,0,-18.5-4.4])
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{
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cube([22.5+0.5,12+0.5,33.5-5.5],center=true); // body
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translate([0,0,28/2-3]) cube([32.5,12.5,2.5],center=true); // bracket
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translate([-6,0,-14]) cylinder(r=2.8,h=35.5+4); // bearing
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translate([27.5/2,0,5]) cylinder(r=inner_screw,h=10); // screw
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translate([-27.5/2,0,5]) cylinder(r=inner_screw,h=10); // screw
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translate([-22.5/2-0.6,0,0]) cube([1.2,4,28],center=true); // cable
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}
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}
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n1 = 33; //red gear number of teeth
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n1_2 = 13;
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n2 = 33; //green gear
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n3 = 5; //blue gear
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n4 = 20; //orange gear
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n5 = 30; //gray rack
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n_servo = 20;
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mm_per_tooth = 3.33; //all meshing gears need the same mm_per_tooth (and the same pressure_angle)
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thickness = 5;
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thickness_servo = 2.8;
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hole = 5+0.6;
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hole_screw = 2.1;
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kopf_screw = 1.5;
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height = 10;
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twist = 8;
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d1 =pitch_radius(mm_per_tooth,n1);
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d12=pitch_radius(mm_per_tooth,n1_2) + pitch_radius(mm_per_tooth,n2);
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d13=pitch_radius(mm_per_tooth,n1) + pitch_radius(mm_per_tooth,n3);
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d14=pitch_radius(mm_per_tooth,n1) + pitch_radius(mm_per_tooth,n4);
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mink=1;
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module servo_gear() {
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translate([ 0, 0, 0]) rotate([0,0, $t*360/n1])
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difference() {
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gear(mm_per_tooth,n2,thickness,hole_screw,twist,108); // servo gear
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cylinder(r=4.5/2, h=kopf_screw*2, center=true, $fn=100);
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//gear(0.77,n_servo,thickness_servo*2,0,0,108); // servo gear
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};
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echo("pitch radius of servo bearing is: ", pitch_radius(0.79, n_servo));
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}
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module rod() {
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assign(pi = 3.1415926)
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translate([mm_per_tooth*(-floor(n5/2)-floor(n1/2)+$t+n1/(3/2)-1/2)+100, +d1+0.4, 0]) rotate([0,0,180]) {
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rack(mm_per_tooth,n5,thickness,height, -twist*atan((2*pi*pitch_radius(mm_per_tooth,n1))/mm_per_tooth));
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}
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}
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module rod_pusher() {
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translate([mm_per_tooth*(-floor(n5/2)-floor(n1/2)+$t+n1/(3/2)-1/2)+100, +d1+0.4, 0]) rotate([0,0,180])
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difference() {
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minkowski() {
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translate([0,2,0])rotate([90,-90,0])linear_extrude(height=9.7)polygon([[0,0],[-15,10],[15,10]]);
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cylinder(r=mink,h=0.2,$fn=50);
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rotate([90,0,0])cylinder(r=mink,h=0.2,$fn=50);
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}
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translate([-3,-8/2-1.6,0])cube([12,10,thickness+0.4],center=true);
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}
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}
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module slid() {
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assign(pi = 3.1415926)
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// translate([mm_per_tooth*(-floor(n5/2)-floor(n1/2)+$t+n1/(3/2)-1/2)+100, +d1+0.4+thickness, -thickness/2]) rotate([0,0,180]) color([0.75,0.75,0.75])
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translate([-50,d1-0.6+mink,-(thickness+1.4)/2-mink])
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cube([mm_per_tooth*n5,height+0.4,thickness+1.4+2*mink]);
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}
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module base() {
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x = 40;
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y = 10;
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z = 20;
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mount_width = 8;
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y_offset = 20;
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x_offset = 6;
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translate([-x/2+x_offset,d1+1,-z+thickness])
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cube([x,y,z]);
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translate([-x/2+x_offset,-5,-10.6-mount_width+mink])
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cube([x,d1+5+1+y,mount_width-2.4*mink]);
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translate([-x/2+x_offset-12/2,d1+1+y-0.2,-z/2])
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cube([x+12,0.2,4]);
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}
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module fasteners() {
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translate([-40/2+1,d1+1+10+1,-15/2]) rotate([90,0,0])
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cylinder(r=inner_screw,h=5,$fn=50);
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translate([40*0.75+1,d1+1+10+1,-15/2]) rotate([90,0,0])
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cylinder(r=inner_screw,h=5,$fn=50);
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}
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%servo();
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%color([0.75,1.00,0.75]) servo_gear();
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%color([1.00,0.50,0.50]) rod();
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%color([1.00,0.50,0.50]) rod_pusher();
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difference() {
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minkowski() {
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difference() {
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base();
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slid();
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}
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cylinder(r=mink,h=0.2,$fn=50);
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rotate([90,0,0])cylinder(r=mink,h=0.2,$fn=50);
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}
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servo();
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cylinder(r=d1+3,h=thickness+3,center=true,$fn=50);
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fasteners();
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}
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