Involute gear generator

The involute is the curve a taut string traces as it unwinds from a circle, the base circle. Two gears with involute flanks turn at a constant speed ratio even when their center distance is slightly off, which is why most gears have them. The generator draws the gear from the sizes a gear is ordered or made by, and gives the measures to check a cut gear against.

The teeth

The module m sets the size of the teeth, and with the number of teeth z it gives the pitch diameter d = m z. Two gears mesh when they have the same module and pressure angle. The pressure angle, usually 20°, sets the base circle d_b = d cos α. The tooth has an addendum of 1 m above the pitch circle and a dedendum of 1.25 m below it, so the tip diameter is d + 2 m and the root diameter d − 2.5 m. The root fillet joins the flanks to the root, 0.38 m by default.

Profile shift

A profile shift x moves the tooth outward by x m, as a cutting tool set further from the gear would. The tip and root diameters grow by 2 x m, and the teeth get thicker at the pitch circle, s = m (π/2 + 2 x tan α). A positive shift strengthens small gears and avoids undercut, and too much of it makes the teeth pointed, which the generator reports.

Gear data

The span over k teeth, W = m cos α [π (k − 0.5) + z inv α] + 2 x m sin α, is the distance a caliper measures across k teeth, touching two flanks. It is the usual check of a cut gear, since it does not depend on the tip diameter. The generator takes k so that the caliper touches the flanks near the pitch circle.

Racks and ring gears

A rack is a gear of infinite radius. Its flanks are straight, inclined at the pressure angle, and its teeth stand one circular pitch, π m, apart, with the same addendum, dedendum and root fillet. The number of teeth sets its length, and the height from its back to the pitch line its body. A ring gear has its teeth on the inside. Its tooth spaces have the shape of a gear's teeth with the addendum and the dedendum swapped, so its root diameter is d + 2.5 m and its tip diameter d − 2 m, and the rim thickness sets its outside diameter. The tips must stay outside the point where the line of action touches the base circle of the pinion that runs in the ring, else they cut into its flanks. The generator therefore takes the pinion's number of teeth and shortens the ring's teeth where that pinion needs it. Below about 10 teeth of difference, the tips of ring gear and pinion collide as the teeth come into mesh, and the generator says so.

Bore and keyway

A bore diameter above 0 cuts the bore, and a keyway width and depth cut a keyway into its top. The depth t2 counts from the bore's edge, as keyway tables give it for the hub.

1. Load

   The page loads Python and planex2d into this tab. Your browser keeps both in its cache, so later visits start faster.

2. Draw

   planex2d draws the gear for the settings and redraws it when one changes.

3. Save

   The gear is saved as DXF, STEP or PDF, made in the tab as well.

The flanks

Each flank is the involute of the base circle as an exact parametric curve, from where it leaves the base circle, or the root circle when that lies further out, to the tip circle. Where the root circle lies inside the base circle, a radial line joins the flank to the root, as on most gear drawings. A cutting tool would undercut the flank there instead, below about 17 teeth at 20° without profile shift, and the generator says so. A ring gear is a ring minus a gear whose teeth have the shape of its tooth spaces. Below about 34 teeth at 20°, its tip circle lies inside the base circle, and the tips of its teeth end in radial lines. Where the fillet asked for does not fit between that line and the root, a smaller one is drawn, and the generator says so.

The files

The DXF file holds the outline in millimeters, a gear's flanks as splines and the tip, root and fillets as arcs. The STEP file holds the gear as a flat face in millimeters, which a CAD system extrudes to the face width. The PDF shows the gear on an A4 sheet at true size when it fits, else at the largest standard scale that does, with its gear data and a line of 50 mm to check the print. The CSV file lists the gear data, a row per quantity with its value and unit.

The same gear in Python

The generator runs the planex2d library, and the same gear takes a few lines of Python with it. The spur gear practice drawing builds it step by step, and in the example Involute spur gears under load, netgen meshes a pair and NGSolve computes the stress under load.