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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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assign(pi = 3.1415926) |
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assign(a = mm_per_tooth / pi) //addendum |
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assign(t = a*tan(pressure_angle)) //tooth side is tilted so top/bottom corners move this amount |
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assign(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+2)*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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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 = 50; //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 = 3; |
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hole = 5+0.2; |
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hole_screw = 2; |
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kopf_screw = 1.5; |
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height = 12; |
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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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/* |
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translate([ 0, 0, 0]) rotate([0,0, $t*360/n1]) color([1.00,0.75,0.75]) gear(mm_per_tooth,n1,thickness,hole, twist); |
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echo("pitch_radius of n1 is: ", d1); |
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translate([ 0, 0, thickness]) rotate([0,0, $t*360/n1]) color([1.00,0.75,0.75]) gear(mm_per_tooth,n1_2,thickness,hole, twist); |
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difference() { |
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translate([ 0, d12, thickness]) rotate([0,0,-($t+n2/2-0*n1+1/2)*360/n2]) color([0.75,1.00,0.75]) gear(mm_per_tooth,n2,thickness,hole_screw,twist,108); // servo gear |
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translate([ 0, d12, (thickness)/2]) color([0.75,1.00,0.75]) cylinder(r=4.5/2, h=kopf_screw*2, center=true, $fn=100); |
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translate([ 0, d12, thickness*1.5]) rotate([0,0,-($t+n2/2-0*n1+1/2)*360/n2]) color([0.75,1.00,0.75]) gear(0.79,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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translate([mm_per_tooth*(-floor(n5/2)-floor(n1/2)+$t+n1/(3/2)-1/2)+100, +d1+0.0, 0]) rotate([0,0,180]) color([0.75,0.75,0.75]) rack(mm_per_tooth,n5,thickness,height, twist); |
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