Cycloidal Disc Reducer

What is modelled

A disc with N-1 = 10 lobes is carried on an eccentric of radius e and rolls without slipping against N = 11 fixed pin rollers on a circle of radius Rp. Because the disc has one lobe fewer than there are pins, every input turn makes the disc fall back by one lobe: the disc (and the output carrier that picks up its motion through the holes) turns once per ten input turns in the opposite direction.

Equations

• Disc profile (pin-circle form of the epitrochoid envelope): x = Rp cosφ - rr cos(φ + ψ) - e cos(Nφ), y = -Rp sinφ + rr sin(φ + ψ) + e sin(Nφ), with ψ = atan[ sin((1-N)φ) / (Rp/(eN) - cos((1-N)φ)) ].
• Disc centre e(cos θ, sin θ), disc spin -θ/(N-1), speed ratio -1/(N-1).
• Rigid-body load share: the contact normals pass through the instantaneous centre N e (cos θ, sin θ); the force on pin i is proportional to its positive lever arm d_i about the disc centre and Σ Fi di = T.
• Ideal power balance: Tin = Tout / (10 η).

Checks run in the page

• All 11 rollers touch the drawn profile (largest gap, in mm).
• A numerical search for the spin angle that keeps every pin tangent returns the ratio -1/10 independently of the formula.
• The pin forces add up to the output torque and 3 to 8 pins carry load at any angle.

Honest simplifications

Rigid bodies, no clearance, no friction in the contacts (efficiency is a single number), no roller deflection (real load share is spread over more pins by Hertzian contact), and one disc pair shown as a single disc. The 3D page models the same kinematics with two discs 180 degrees apart.

• Slow the input to 30 rpm and follow one marked pin: the disc advances one lobe per input turn.
• Raise the load torque and watch which pins glow red; about half of them share the torque.
• Push the eccentricity near its limit and see on the lower-right graph how the peak pin force changes, then watch the warning when the profile starts to loop.