Involute gears

involute.scad

// Parametric Involute Bevel and Spur Gears by GregFrost
// It is licensed under the Creative Commons - GNU GPL license.
// © 2010 by GregFrost
// http://www.thingiverse.com/thing:3575

// Simple Test:
//gear (circular_pitch=700,
//	gear_thickness = 12,
//	rim_thickness = 15,
//	hub_thickness = 17,
//	circles=8);

//Complex Spur Gear Test:
//test_gears ();

// Meshing Double Helix:
//meshing_double_helix ();

// Demonstrate the backlash option for Spur gears.
//test_backlash ();

// Demonstrate how to make meshing bevel gears.
bevel_gear_pair ();

pi=3.1415926535897932384626433832795;

//==================================================
// Bevel Gears:
// Two gears with the same cone distance, circular pitch (measured at the cone distance) 
// and pressure angle will mesh.

module bevel_gear_pair (
	gear1_teeth = 41,
	gear2_teeth = 7,
	axis_angle = 90,
	outside_circular_pitch=1000)
{
	outside_pitch_radius1 = gear1_teeth * outside_circular_pitch / 360;
	outside_pitch_radius2 = gear2_teeth * outside_circular_pitch / 360;
	pitch_apex1=outside_pitch_radius2 * sin (axis_angle) + 
		(outside_pitch_radius2 * cos (axis_angle) + outside_pitch_radius1) / tan (axis_angle);
	cone_distance = sqrt (pow (pitch_apex1, 2) + pow (outside_pitch_radius1, 2));
	pitch_apex2 = sqrt (pow (cone_distance, 2) - pow (outside_pitch_radius2, 2));
	echo ("cone_distance", cone_distance);
	pitch_angle1 = asin (outside_pitch_radius1 / cone_distance);
	pitch_angle2 = asin (outside_pitch_radius2 / cone_distance);
	echo ("pitch_angle1, pitch_angle2", pitch_angle1, pitch_angle2);
	echo ("pitch_angle1 + pitch_angle2", pitch_angle1 + pitch_angle2);

	rotate([0,0,90])
	translate ([0,0,pitch_apex1+20])
	{
		translate([0,0,-pitch_apex1])
		bevel_gear (
			number_of_teeth=gear1_teeth,
			cone_distance=cone_distance,
			pressure_angle=30,
			outside_circular_pitch=outside_circular_pitch);
	
		rotate([0,-(pitch_angle1+pitch_angle2),0])
		translate([0,0,-pitch_apex2])
		bevel_gear (
			number_of_teeth=gear2_teeth,
			cone_distance=cone_distance,
			pressure_angle=30,
			outside_circular_pitch=outside_circular_pitch);
	}
}

//Bevel Gear Finishing Options:
bevel_gear_flat = 0;
bevel_gear_back_cone = 1;

module bevel_gear (
	number_of_teeth=11,
	cone_distance=100,
	face_width=20,
	outside_circular_pitch=1000,
	pressure_angle=30,
	clearance = 0.2,
	bore_diameter=5,
	gear_thickness = 15,
	backlash = 0,
	involute_facets=0,
	finish = -1)
{
	echo ("bevel_gear",
		"teeth", number_of_teeth,
		"cone distance", cone_distance,
		face_width,
		outside_circular_pitch,
		pressure_angle,
		clearance,
		bore_diameter,
		involute_facets,
		finish);

	// Pitch diameter: Diameter of pitch circle at the fat end of the gear.
	outside_pitch_diameter  =  number_of_teeth * outside_circular_pitch / 180;
	outside_pitch_radius = outside_pitch_diameter / 2;

	// The height of the pitch apex.
	pitch_apex = sqrt (pow (cone_distance, 2) - pow (outside_pitch_radius, 2));
	pitch_angle = asin (outside_pitch_radius/cone_distance);

	echo ("Num Teeth:", number_of_teeth, " Pitch Angle:", pitch_angle);

	finish = (finish != -1) ? finish : (pitch_angle < 45) ? bevel_gear_flat : bevel_gear_back_cone;

	apex_to_apex=cone_distance / cos (pitch_angle);
	back_cone_radius = apex_to_apex * sin (pitch_angle);

	// Calculate and display the pitch angle. This is needed to determine the angle to mount two meshing cone gears.

	// Base Circle for forming the involute teeth shape.
	base_radius = back_cone_radius * cos (pressure_angle);	

	// Diametrial pitch: Number of teeth per unit length.
	pitch_diametrial = number_of_teeth / outside_pitch_diameter;

	// Addendum: Radial distance from pitch circle to outside circle.
	addendum = 1 / pitch_diametrial;
	// Outer Circle
	outer_radius = back_cone_radius + addendum;

	// Dedendum: Radial distance from pitch circle to root diameter
	dedendum = addendum + clearance;
	dedendum_angle = atan (dedendum / cone_distance);
	root_angle = pitch_angle - dedendum_angle;

	root_cone_full_radius = tan (root_angle)*apex_to_apex;
	back_cone_full_radius=apex_to_apex / tan (pitch_angle);

	back_cone_end_radius = 
		outside_pitch_radius - 
		dedendum * cos (pitch_angle) - 
		gear_thickness / tan (pitch_angle);
	back_cone_descent = dedendum * sin (pitch_angle) + gear_thickness;

	// Root diameter: Diameter of bottom of tooth spaces.
	root_radius = back_cone_radius - dedendum;

	half_tooth_thickness = outside_pitch_radius * sin (360 / (4 * number_of_teeth)) - backlash / 4;
	half_thick_angle = asin (half_tooth_thickness / back_cone_radius);

	face_cone_height = apex_to_apex-face_width / cos (pitch_angle);
	face_cone_full_radius = face_cone_height / tan (pitch_angle);
	face_cone_descent = dedendum * sin (pitch_angle);
	face_cone_end_radius = 
		outside_pitch_radius -
		face_width / sin (pitch_angle) - 
		face_cone_descent / tan (pitch_angle);

	// For the bevel_gear_flat finish option, calculate the height of a cube to select the portion of the gear that includes the full pitch face.
	bevel_gear_flat_height = pitch_apex - (cone_distance - face_width) * cos (pitch_angle);

//	translate([0,0,-pitch_apex])
	difference ()
	{
		intersection ()
		{
			union()
			{
				rotate (half_thick_angle)
				translate ([0,0,pitch_apex-apex_to_apex])
				cylinder ($fn=number_of_teeth*2, r1=root_cone_full_radius,r2=0,h=apex_to_apex);
				for (i = [1:number_of_teeth])
//				for (i = [1:1])
				{
					rotate ([0,0,i*360/number_of_teeth])
					{
						involute_bevel_gear_tooth (
							back_cone_radius = back_cone_radius,
							root_radius = root_radius,
							base_radius = base_radius,
							outer_radius = outer_radius,
							pitch_apex = pitch_apex,
							cone_distance = cone_distance,
							half_thick_angle = half_thick_angle,
							involute_facets = involute_facets);
					}
				}
			}

			if (finish == bevel_gear_back_cone)
			{
				translate ([0,0,-back_cone_descent])
				cylinder (
					$fn=number_of_teeth*2, 
					r1=back_cone_end_radius,
					r2=back_cone_full_radius*2,
					h=apex_to_apex + back_cone_descent);
			}
			else
			{
				translate ([-1.5*outside_pitch_radius,-1.5*outside_pitch_radius,0])
				cube ([3*outside_pitch_radius,
					3*outside_pitch_radius,
					bevel_gear_flat_height]);
			}
		}
		
		if (finish == bevel_gear_back_cone)
		{
			translate ([0,0,-face_cone_descent])
			cylinder (
				r1=face_cone_end_radius,
				r2=face_cone_full_radius * 2,
				h=face_cone_height + face_cone_descent+pitch_apex);
		}

		translate ([0,0,pitch_apex - apex_to_apex])
		cylinder (r=bore_diameter/2,h=apex_to_apex);
	}	
}

module involute_bevel_gear_tooth (
	back_cone_radius,
	root_radius,
	base_radius,
	outer_radius,
	pitch_apex,
	cone_distance,
	half_thick_angle,
	involute_facets)
{
//	echo ("involute_bevel_gear_tooth",
//		back_cone_radius,
//		root_radius,
//		base_radius,
//		outer_radius,
//		pitch_apex,
//		cone_distance,
//		half_thick_angle);

	min_radius = max (base_radius*2,root_radius*2);

	pitch_point = 
		involute (
			base_radius*2, 
			involute_intersect_angle (base_radius*2, back_cone_radius*2));
	pitch_angle = atan2 (pitch_point[1], pitch_point[0]);
	centre_angle = pitch_angle + half_thick_angle;

	start_angle = involute_intersect_angle (base_radius*2, min_radius);
	stop_angle = involute_intersect_angle (base_radius*2, outer_radius*2);

	res=(involute_facets!=0)?involute_facets:($fn==0)?5:$fn/4;

	translate ([0,0,pitch_apex])
	rotate ([0,-atan(back_cone_radius/cone_distance),0])
	translate ([-back_cone_radius*2,0,-cone_distance*2])
	union ()
	{
		for (i=[1:res])
		{
			point1=
					involute (base_radius*2,start_angle+(stop_angle - start_angle)*(i-1)/res);
			point2=
					involute (base_radius*2,start_angle+(stop_angle - start_angle)*(i)/res);
			{
				side1_point1 = rotate_point (centre_angle, point1);
				side1_point2 = rotate_point (centre_angle, point2);
				side2_point1 = mirror_point (rotate_point (centre_angle, point1));
					side2_point2 = mirror_point (rotate_point (centre_angle, point2));
				{
					polyhedron (
						points=[
							[back_cone_radius*2+0.1,0,cone_distance*2],
							[side1_point1[0],side1_point1[1],0],
							[side1_point2[0],side1_point2[1],0],
							[side2_point2[0],side2_point2[1],0],
							[side2_point1[0],side2_point1[1],0],
							[0.1,0,0]],
						faces=[[0,2,1],[0,3,2],[0,4,3],[0,1,5],[1,2,5],[2,3,5],[3,4,5],[0,5,4]]);
				}
			}
		}
	}
}

module gear (
	number_of_teeth=15,
	circular_pitch=false, diametral_pitch=false,
	pressure_angle=28,
	clearance = 0.2,
	gear_thickness=5,
	rim_thickness=8,
	rim_width=5,
	hub_thickness=10,
	hub_diameter=15,
	bore_diameter=5,
	circles=0,
	backlash=0,
	twist=0,
	involute_facets=0)
{
	if (circular_pitch==false && diametral_pitch==false) 
		echo("MCAD ERROR: gear module needs either a diametral_pitch or circular_pitch");

	//Convert diametrial pitch to our native circular pitch
	circular_pitch = (circular_pitch!=false?circular_pitch:180/diametral_pitch);

	// Pitch diameter: Diameter of pitch circle.
	pitch_diameter  =  number_of_teeth * circular_pitch / 180;
	pitch_radius = pitch_diameter/2;
	echo ("Teeth:", number_of_teeth, " Pitch radius:", pitch_radius);

	// Base Circle
	base_radius = pitch_radius*cos(pressure_angle);

	// Diametrial pitch: Number of teeth per unit length.
	pitch_diametrial = number_of_teeth / pitch_diameter;

	// Addendum: Radial distance from pitch circle to outside circle.
	addendum = 1/pitch_diametrial;

	//Outer Circle
	outer_radius = pitch_radius+addendum;

	// Dedendum: Radial distance from pitch circle to root diameter
	dedendum = addendum + clearance;

	// Root diameter: Diameter of bottom of tooth spaces.
	root_radius = pitch_radius-dedendum;
	backlash_angle = backlash / pitch_radius * 180 / pi;
	half_thick_angle = (360 / number_of_teeth - backlash_angle) / 4;

	// Variables controlling the rim.
	rim_radius = root_radius - rim_width;

	// Variables controlling the circular holes in the gear.
	circle_orbit_diameter=hub_diameter/2+rim_radius;
	circle_orbit_curcumference=pi*circle_orbit_diameter;

	// Limit the circle size to 90% of the gear face.
	circle_diameter=
		min (
			0.70*circle_orbit_curcumference/circles,
			(rim_radius-hub_diameter/2)*0.9);

	difference ()
	{
		union ()
		{
			difference ()
			{
				linear_extrude (height=rim_thickness, convexity=10, twist=twist)
				gear_shape (
					number_of_teeth,
					pitch_radius = pitch_radius,
					root_radius = root_radius,
					base_radius = base_radius,
					outer_radius = outer_radius,
					half_thick_angle = half_thick_angle,
					involute_facets=involute_facets);

				if (gear_thickness < rim_thickness)
					translate ([0,0,gear_thickness])
					cylinder (r=rim_radius,h=rim_thickness-gear_thickness+1);
			}
			if (gear_thickness > rim_thickness)
				cylinder (r=rim_radius,h=gear_thickness);
			if (hub_thickness > gear_thickness)
				translate ([0,0,gear_thickness])
				cylinder (r=hub_diameter/2,h=hub_thickness-gear_thickness);
		}
		translate ([0,0,-1])
		cylinder (
			r=bore_diameter/2,
			h=2+max(rim_thickness,hub_thickness,gear_thickness));
		if (circles>0)
		{
			for(i=[0:circles-1])	
				rotate([0,0,i*360/circles])
				translate([circle_orbit_diameter/2,0,-1])
				cylinder(r=circle_diameter/2,h=max(gear_thickness,rim_thickness)+3);
		}
	}
}

module gear_shape (
	number_of_teeth,
	pitch_radius,
	root_radius,
	base_radius,
	outer_radius,
	half_thick_angle,
	involute_facets)
{
	union()
	{
		rotate (half_thick_angle) circle ($fn=number_of_teeth*2, r=root_radius);

		for (i = [1:number_of_teeth])
		{
			rotate ([0,0,i*360/number_of_teeth])
			{
				involute_gear_tooth (
					pitch_radius = pitch_radius,
					root_radius = root_radius,
					base_radius = base_radius,
					outer_radius = outer_radius,
					half_thick_angle = half_thick_angle,
					involute_facets=involute_facets);
			}
		}
	}
}

module involute_gear_tooth (
	pitch_radius,
	root_radius,
	base_radius,
	outer_radius,
	half_thick_angle,
	involute_facets)
{
	min_radius = max (base_radius,root_radius);

	pitch_point = involute (base_radius, involute_intersect_angle (base_radius, pitch_radius));
	pitch_angle = atan2 (pitch_point[1], pitch_point[0]);
	centre_angle = pitch_angle + half_thick_angle;

	start_angle = involute_intersect_angle (base_radius, min_radius);
	stop_angle = involute_intersect_angle (base_radius, outer_radius);

	res=(involute_facets!=0)?involute_facets:($fn==0)?5:$fn/4;

	union ()
	{
		for (i=[1:res])
		assign (
			point1=involute (base_radius,start_angle+(stop_angle - start_angle)*(i-1)/res),
			point2=involute (base_radius,start_angle+(stop_angle - start_angle)*i/res))
		{
			assign (
				side1_point1=rotate_point (centre_angle, point1),
				side1_point2=rotate_point (centre_angle, point2),
				side2_point1=mirror_point (rotate_point (centre_angle, point1)),
				side2_point2=mirror_point (rotate_point (centre_angle, point2)))
			{
				polygon (
					points=[[0,0],side1_point1,side1_point2,side2_point2,side2_point1],
					paths=[[0,1,2,3,4,0]]);
			}
		}
	}
}

// Mathematical Functions
//===============

// Finds the angle of the involute about the base radius at the given distance (radius) from it's center.
//source: http://www.mathhelpforum.com/math-help/geometry/136011-circle-involute-solving-y-any-given-x.html

function involute_intersect_angle (base_radius, radius) = sqrt (pow (radius/base_radius, 2) - 1) * 180 / pi;

// Calculate the involute position for a given base radius and involute angle.

function rotated_involute (rotate, base_radius, involute_angle) = 
[
	cos (rotate) * involute (base_radius, involute_angle)[0] + sin (rotate) * involute (base_radius, involute_angle)[1],
	cos (rotate) * involute (base_radius, involute_angle)[1] - sin (rotate) * involute (base_radius, involute_angle)[0]
];

function mirror_point (coord) = 
[
	coord[0], 
	-coord[1]
];

function rotate_point (rotate, coord) =
[
	cos (rotate) * coord[0] + sin (rotate) * coord[1],
	cos (rotate) * coord[1] - sin (rotate) * coord[0]
];

function involute (base_radius, involute_angle) = 
[
	base_radius*(cos (involute_angle) + involute_angle*pi/180*sin (involute_angle)),
	base_radius*(sin (involute_angle) - involute_angle*pi/180*cos (involute_angle)),
];


// Test Cases
//===============

module test_gears()
{
	translate([17,-15])
	{
		gear (number_of_teeth=17,
			circular_pitch=500,
			circles=8);
	
		rotate ([0,0,360*4/17])
		translate ([39.088888,0,0])
		{
			gear (number_of_teeth=11,
				circular_pitch=500,
				hub_diameter=0,
				rim_width=65);
			translate ([0,0,8])
			{
				gear (number_of_teeth=6,
					circular_pitch=300,
					hub_diameter=0,
					rim_width=5,
					rim_thickness=6,
					pressure_angle=31);
				rotate ([0,0,360*5/6])
				translate ([22.5,0,1])
				gear (number_of_teeth=21,
					circular_pitch=300,
					bore_diameter=2,
					hub_diameter=4,
					rim_width=1,
					hub_thickness=4,
					rim_thickness=4,
					gear_thickness=3,
					pressure_angle=31);
			}
		}

		translate ([-61.1111111,0,0])
		{
			gear (number_of_teeth=27,
				circular_pitch=500,
				circles=5,
				hub_diameter=2*8.88888889);

			translate ([0,0,10])
			{
				gear (
					number_of_teeth=14,
					circular_pitch=200,
					pressure_angle=5,
					clearance = 0.2,
					gear_thickness = 10,
					rim_thickness = 10,
					rim_width = 15,
					bore_diameter=5,
					circles=0);
				translate ([13.8888888,0,1])
				gear (
					number_of_teeth=11,
					circular_pitch=200,
					pressure_angle=5,
					clearance = 0.2,
					gear_thickness = 10,
					rim_thickness = 10,
					rim_width = 15,
					hub_thickness = 20,
					hub_diameter=2*7.222222,
					bore_diameter=5,
					circles=0);
			}
		}
	
		rotate ([0,0,360*-5/17])
		translate ([44.444444444,0,0])
		gear (number_of_teeth=15,
			circular_pitch=500,
			hub_diameter=10,
			rim_width=5,
			rim_thickness=5,
			gear_thickness=4,
			hub_thickness=6,
			circles=9);
	
		rotate ([0,0,360*-1/17])
		translate ([30.5555555,0,-1])
		gear (number_of_teeth=5,
			circular_pitch=500,
			hub_diameter=0,
			rim_width=5,
			rim_thickness=10);
	}
}

module meshing_double_helix ()
{
	test_double_helix_gear ();
	
	mirror ([0,1,0])
	translate ([58.33333333,0,0])
	test_double_helix_gear (teeth=13,circles=6);
}

module test_double_helix_gear (
	teeth=17,
	circles=8)
{
	//double helical gear
	{
		twist=200;
		height=20;
		pressure_angle=30;

		gear (number_of_teeth=teeth,
			circular_pitch=700,
			pressure_angle=pressure_angle,
			clearance = 0.2,
			gear_thickness = height/2*0.5,
			rim_thickness = height/2,
			rim_width = 5,
			hub_thickness = height/2*1.2,
			hub_diameter=15,
			bore_diameter=5,
			circles=circles,
			twist=twist/teeth);
		mirror([0,0,1])
		gear (number_of_teeth=teeth,
			circular_pitch=700,
			pressure_angle=pressure_angle,
			clearance = 0.2,
			gear_thickness = height/2,
			rim_thickness = height/2,
			rim_width = 5,
			hub_thickness = height/2,
			hub_diameter=15,
			bore_diameter=5,
			circles=circles,
			twist=twist/teeth);
	}
}

module test_backlash ()
{
	backlash = 2;
	teeth = 15;

	translate ([-29.166666,0,0])
	{
		translate ([58.3333333,0,0])
		rotate ([0,0,-360/teeth/4])
		gear (
			number_of_teeth = teeth,
			circular_pitch=700,
			gear_thickness = 12,
			rim_thickness = 15,
			rim_width = 5,
			hub_thickness = 17,
			hub_diameter=15,
			bore_diameter=5,
			backlash = 2,
			circles=8);
		
		rotate ([0,0,360/teeth/4])
		gear (
			number_of_teeth = teeth,
			circular_pitch=700,
			gear_thickness = 12,
			rim_thickness = 15,
			rim_width = 5,
			hub_thickness = 17,
			hub_diameter=15,
			bore_diameter=5,
			backlash = 2,
			circles=8);
	}

	color([0,0,128,0.5])
	translate([0,0,-5])
	cylinder ($fn=20,r=backlash / 4,h=25);
}