Cycloidal Reducer

What is modelled

A disc with Z lobes sits on an eccentric and meshes with N = Z + 1 pins fixed on a ring. As the input shaft makes one turn, the disc centre orbits once and the disc is forced to turn back by one lobe, 1/Z of a revolution, so the output turns at 1/Z of the input speed in the opposite direction. A second disc 180 degrees away balances the first.

Equations

• Disc profile (epitrochoid offset by the pin radius) x = R cosθ − rp cos(θ+ψ) − e cos(Nθ), y = −R sinθ + rp sin(θ+ψ) + e sin(Nθ), ψ = atan[ sin((1−N)θ) / (R/(eN) − cos((1−N)θ)) ].
• Ratio i = Z = N − 1; Tout = η i Tin.
• Pins on one side carry F = Fmax sinφ, so Fmax = Tout/(R Σ sin²φ) = 4 Tout/(N R).
• Shaking force of one disc m e ω²; two opposite discs cancel it and leave a couple.

Checks run in the page

• The disc radius running between R − r_p − e and R − r_p + e.
• The profile mapping onto itself when turned by 360 / Z degrees.
• The closest approach of every pin to the profile at four input angles.
• The sum of sin² over the loaded pins and the net force of two discs.

Honest simplifications

Contact is treated as perfectly rigid and frictionless with all loaded pins sharing the force in proportion to their lever arm (real pins share less evenly because of clearance and Hertz deformation), the efficiency is a fixed number rather than a result of friction, and the disc holes, output pins and bearings are only drawn, not loaded.

• Press Many lobes, high ratio and watch the reduction climb to 31:1 while the pins shrink.
• Raise the eccentricity: the disc lobes get deeper and the pin loads shift, until the geometry would undercut.
• Raise the speed and the disc mass and compare the shaking force of one disc with two discs in the last plot.