Rack profile calculation is incorrect

[tombsar]

The rack looks correct with the default pressure angle of 28 degrees, but changing this to a lower value produces an incorrect result with intersection between the rack and meshed gears.

This can be reproduced with the following example (replacing the example gears in the code):

n1 = 11; //red gear number of teeth
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);

translate([ 0,    0, 0]) rotate([0,0, $t*360/n1])                 color([1.00,0.75,0.75]) gear(mm_per_tooth,n1,thickness,hole,pressure_angle=14.5);
translate([(-floor(n5/2)-floor(n1/2)+$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,pressure_angle=14.5);

[north90]

I noticed the same problem.

change t = a * cos(pressure_angle)-mm_per_tooth/10; (line119?)
to t = a * sin(pressure_angle)

[jonathanperret]

I hit this as well today. But I think the correct formula would be

t = a * tan(pressure_angle);

Since we are computing (t) how long the bottom side of a right triangle (of height a) needs to be so that the hypotenuse makes an angle of pressure_angle with the vertical side.

[alastaira]

Agreed, the current calculation is definitely incorrect. Can't get my head around whether sin() or tan() is the correct alternative though!