A cycloidal drive turns a small wobble and a one-tooth count mismatch into a large, smooth, backlash-resistant reduction. This guide walks through the design process as a chain of decisions — each number you choose hands you the next. Finally we arrive at the governing equations, after which the built-in interactive rotor equation generator creates the parametric equations needed based on your parameters.
Four parts do all the work. An input shaft carries an eccentric cam — a journal offset from the axis — that rides inside the cycloidal disc (the rotor). Instead of spinning the rotor, the cam pushes it sideways so it orbits the centre. Around the rotor sits a ring of fixed roller pins, and the rotor's lobes press against them. Finally, a set of output pins passes through oversized holes in the rotor and carries its motion to the output flange.
The trick is in the counts. The rotor always has one fewer lobe than there are ring pins. As the cam drives the rotor around its orbit, the lobes can never line up with the pins all the way around — so each full turn of the input forces the rotor to creep backward by exactly one lobe. That backward creep is the output. Because one input turn advances the rotor by a single lobe — that is, 1 / (N − 1) of a revolution — the output turns once for every N − 1 turns of the input. Ten lobes give a clean 10 : 1, the output running opposite to the input.
Two things fall out of this geometry for free. At any instant roughly half the lobes are engaged at once, sharing the load — which is why these drives are so compact yet handle high torque and sudden shocks. And because contact is rolling rather than sliding, a well-made cycloidal drive runs with very little backlash.
The whole design is a chain. You make one decision, and the math hands you the next. Here is that chain end to end, in plain terms.
Lobes = the reduction; pins = lobes + 1. Start with the job. The reduction your application needs fixes both tooth counts at once: a ratio of i means i lobes running against i + 1 ring pins. A 20 : 1 drive is a 20-lobe rotor and 21 rollers. Nothing else is decided yet, but the counts are now locked.
The pin-circle radius R is the master dimension. Decide how big the drive can be. The rollers sit on a circle, and its radius R effectively sets the outer diameter — a little over 2R once you add the rollers and housing. Pick R from the space you have. Everything downstream is measured against it, which is what lets the rest be calculated rather than guessed.
With R and N fixed, the two shape numbers follow. The eccentricity E sets how deep the lobes are — the biggest single lever on torque capacity — and it must stay below R / N, or the profile develops cusps and self-intersects. A safe starting point is half that limit, E ≈ R / (2N). The roller radius Rr sets the size of each pin; a good range is R / N to R / (1.5 N). Larger rollers spread the contact stress and last longer, but they shallow the lobes and, pushed too far, interfere. Both numbers are computed from R and N.
Four numbers define the curve. With R, Rr, E and N in hand, feed them into the equations in §3 and sweep the parameter t from 0 to 2π to trace the complete lobed outline. Plug your parameters into the interactive generator below and it will output the parametric equations for you to put into your dedicated CAD software. Note the tangent function is labelled atn for SolidWorks; if you are using a different program you may need to change atn to that software's specific function.
Pins through oversized holes carry the slow spin out. Because the rotor is orbiting, each output hole must be larger than its pin by exactly the orbit (one eccentricity), so the wobble is absorbed while the rotation passes through: Rhole = Rpin + E. Space six or eight pins evenly so the load shares and the output runs smoothly.
A single disc shakes; a second one balances it — if you clock it right. Put the second disc on a cam offset 180° from the first so the two orbiting masses cancel. Generate its lobe profile identically to the first — you do not rotate the profile. Instead, offset its output holes by half a lobe pitch:
cams 180° apart for balance · the second rotor's output holes indexed by 180° / (N − 1)
It is the holes that have to rotate, not the lobes. The output pins are shared between both discs, so the hole pattern is the timing reference: offsetting the second disc's holes by 180° / (N − 1) is what forces that disc to sit at the correct angle such that the output holes align and the rotational motion can be captured and transmitted into the output flange. For a 10-lobe drive that offset is 18°.
Finally, the mechanics. The cam journal is offset from the shaft by exactly E; ride each rotor on it through a needle or ball bearing, sized for the full transmitted load since that bearing carries the torque. Assemble the cam, both rotors, the roller ring and the output pins — and every dimension traces back to the two decisions you made first: the reduction and the size.
The rotor edge is a cycloidal curve offset by the roller radius, and it is built from just the four numbers you set — the pin count N, the pin-circle radius R, the eccentricity E, and the roller radius Rr. Sweep t from 0 to 2π to draw one full disc; t is only the sweep variable, not something you choose. For readability the repeated arctangent term is named the contact angle, ψ.
Most CAD packages take an equation-driven curve directly. The single-expression form below is ready to paste, and it uses atn — the arctangent name SolidWorks expects (Fusion 360 and others use atan or arctan; just swap the name). Set the sweep to t = 0 to 2*pi.
Set the parameters and the rotor redraws live against its ring of rollers, auto-scaled to fit. The rotor is defined by four numbers — lobes, R, Rr and E — and the E and Rr sliders are hard-limited to the valid ranges established earlier (E from R/2N to R/N; Rr from R/1.5N to R/N), so neither can be driven outside what the equations allow. Three controls set the output pins (the output circle is capped at R), and one more opens the central bore for the shaft or cam bearing. When it looks right, export the exact rotor you just made as a DXF or CSV, or copy the SolidWorks equation with your numbers already in place.
Four quantities define the rotor — N, R, Rr and E. (The sweep variable t isn't a design value; it just runs 0 → 2π to trace the curve.) What matters in practice is how each one changes the behaviour of the gear.
| Symbol | What it is | How it affects the gear |
|---|---|---|
| N | Number of ring pins (lobes = N − 1) | Sets the reduction (N − 1) and the lobe count. More pins → higher ratio with more, smaller, shallower lobes that share load smoothly; fewer pins → lower ratio with deeper, more aggressive lobes. |
| R | Pin-circle radius | The master size dimension — fixes the outer diameter (≈ 2R). Larger R means a bigger, stronger, higher-torque drive but more bulk and weight. |
| Rr | Roller (ring-pin) radius | Larger rollers spread contact stress and improve durability, but shallow the lobes and reduce usable profile; oversize rollers interfere. Range R/N to R/(1.5·N). |
| E | Eccentricity | The torque lever. Larger E deepens the lobes for more torque capacity and bigger output holes, but raises the pressure angle and contact forces (more stress and vibration). Keep it between R/2N and the R/N limit. |