////////////////////////////////////////////////////////////////////////////////////////////// // 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 // use // ``` ////////////////////////////////////////////////////////////////////////////////////////////// use use include // 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. // valleys = If true, adds valley extentions to the base of the gear tooth. Default: true // Example(2D): // gear_tooth_profile(mm_per_tooth=5, number_of_teeth=20, pressure_angle=20); function _gear_polar(r,theta) = r*[sin(theta), cos(theta)]; //convert polar to cartesian coordinates function _gear_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 _gear_q7(f,r,b,r2,t,s) = _gear_q6(b,s,t,(1-f)*max(b,r)+f*r2); //radius a fraction f up the curved side of the tooth function _gear_q6(b,s,t,d) = _gear_polar(d,s*(_gear_iang(b,d)+t)); //point at radius d on the involute curve function 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, valleys = true ) = let( 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 = -_gear_iang(b, p) - t/2/p/PI*180, //angle to where involute meets base circle on each side of tooth mat = matrix3_scale([1,1/cos(bevelang), 1]) * matrix3_translate([0,-r,0]), points=[ if(valleys) _gear_polar(r, -181/number_of_teeth), _gear_polar(r, r0) [0,0] ] ) pts; 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 ) { path = 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 ); polygon(path); } // 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`. // align = Alignment of the gear. Use the `V_` constants from `constants.scad`. Default: `V_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, align = V_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_align([p, p, thickness], orient, align) { 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); } } } } } } // 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`. // 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); 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, align = V_RIGHT ) { a = adendum(mm_per_tooth); d = dedendum(mm_per_tooth, clearance); xa = a * sin(pressure_angle); xd = d * sin(pressure_angle); orient_and_align([(number_of_teeth-1)*mm_per_tooth, height, thickness], orient, align, orig_orient=ORIENT_X) { 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], ] ); } } } } } } ////////////////////////////////////////////////////////////////////////////////////////////// //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,-($t-n3/4+n1/4+1/2)*360/n3]) color([0.75,0.75,1.00]) gear(mm_per_tooth,n3,thickness,hole); translate([-d14, 0, 0]) rotate([0,0,-($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