Harmonic drives, plainly

A harmonic (strain wave) drive is a gearbox with huge reduction, near-zero backlash, and very few parts. This page explains how it works and what each number in the harmonic maker actually controls.

<the-three-parts/>

Every harmonic drive is three pieces:

• Flexspline, a thin, flexible cup or ring with external teeth. It bends.
• Circular spline, a rigid ring with internal teeth, with two more teeth than the flexspline. It stays round.
• Wave generator, an oval cam (or roller assembly) inside the flexspline that pushes it into an ellipse, so its teeth engage the circular spline only at the two ends of the major axis.

As the wave generator spins, the engagement zone travels around the ring. Because the flexspline has two fewer teeth, one full turn of the wave generator walks the flexspline backward by just two teeth, and that tiny slip per revolution is the huge reduction.

<module-and-teeth/>

Module (m, mm) is tooth size: pitch diameter ÷ tooth count. Bigger module, bigger and stronger teeth. Zf and Zc are the flexspline and circular spline tooth counts. The reduction ratio is:

ratio = Zf / (Zc − Zf)

With the usual two-tooth difference, a 100-tooth flexspline gives 50:1; a 32/34 pancake gives 16:1. Keep Zc − Zf even (almost always 2).

<deflection-ratio/>

The wave generator deflects the flexspline outward by a radial amount w₀ at the major axis. The deflection ratio is w₀ / m. A ratio of 1 means the radius grows by exactly one module, which is what you want, because the two-tooth difference makes the flexspline and circular spline pitch radii differ by exactly one module too. So the teeth mesh fully at the major axis and clear fully at the minor axis.

<the-wave-generator/>

The cam profile isn't a free-form oval. It follows a cosine deflection, r(φ) = R + w₀·cos(2φ), which is the shape a thin ring naturally takes under a smooth two-lobe load. The harmonic maker generates this profile for you: toggle include wave generator and it exports the solid cam as its own DXF, sized to your bore and deflection.

There's one hard limit. To stay convex (no cusp at the minor axis):

R > 5·w₀

where R is the flexspline's neutral radius. Real drives sit far above this (R/w₀ ≈ 40–100); if you ever crowd the limit, the flexspline buckles instead of bending smoothly. If a 34-mm-class cam is awkward to source as a single bearing, build the wave generator from small roller bearings on a carrier instead. The deflecting span is fixed by the bore, but the bearings themselves can be tiny.

<using-the-harmonic-maker/>

• Cup vs pancake: cup is the classic two-part drive; pancake (flat) adds a second, output circular spline.
• Tooth form: a fully-conjugate cycloid (CTP) or the conventional S double-arc.
• Advanced mode exposes addendum, dedendum, clearance, crown and pressure angle if you want to tune the mesh.
• The live readout shows ratio, deflection, teeth in mesh, backlash and flexspline strain as you type.
• Hit copy share link to put the whole design in the URL. Paste it to anyone and they'll see the exact same gear.
• Profiles export as 2D DXF; extrude them to your chosen wall depth in CAD or a slicer.