Planetary Ratio⚠ unverified
Mechanical / Gears · Compute the speed ratio of a planetary gear set with the ring fixed
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| Rs | Rs | — | 1.0 | Number of teeth on the sun gear |
| Rp | Rp | — | 1.0 | Number of teeth on a planet gear |
| Rc | Rc | — | 0.0 | Carrier parameter retained for signature compatibility; a carrier has no tooth count and the value is unused. Default is 0 |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | Zring | — | Dimensionless speed ratio n_sun / n_carrier |
The science & history
Understanding the Parameters
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Sun teeth $Z_{sun}$ — the central driver; a small sun gives a large reduction. It sets the base of both the geometry constraint and the speed ratio.
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Planet teeth $Z_{planet}$ — the planets idle in the ratio (they don't appear in the final speed ratio — they are "idler" gears that reverse nothing net), but they set the ring size through the constraint, since a planet spans from sun to ring. The number of planets (usually 3–5) sets load sharing, not the ratio.
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Ring teeth $Z_{ring}$ — the output of this constraint. Physically it must equal $Z_{sun} + 2Z_{planet}$ so the ring's pitch circle exactly encloses the sun-plus-two-planets diameter. (There is also an assembly condition: $(Z_{sun}+Z_{ring})$ must be divisible by the number of planets for them to space evenly.)
Derivation (Approaching a Proof)
The geometric constraint. Along a line through the centre, the pitch radii stack: from the sun's centre outward you cross the sun's pitch radius $r_s$, then one planet's diameter $2r_p$, to reach the ring's pitch radius $r_{ring}$. Since (for equal module) tooth count is proportional to pitch radius,
$$r_{ring} = r_s + 2 r_p \;\Longrightarrow\; Z_{ring} = Z_{sun} + 2 Z_{planet}.$$
The actual speed ratio (what the name promises) comes from the epicyclic relation, most easily found by Willis's method (add $-\omega_c$ to every member to fix the carrier, analyse as an ordinary gear train, then add $\omega_c$ back). With the ring held fixed and the sun as input, carrier as output:
$$\frac{n_{sun}}{n_{carrier}} = 1 + \frac{Z_{ring}}{Z_{sun}}.$$
So a set with $Z_{sun}=24$, $Z_{ring}=72$ gives a 4:1 reduction ($1 + 72/24$). The planets cancel out of the ratio entirely — they only transmit load and set the ring size. The registry's formula gives the $Z_{ring}$ that this ratio expression needs, i.e. it is the precursor to the ratio, not the ratio itself.
Dimensional check. All tooth counts are dimensionless, so $Z_{ring}$ is a dimensionless count. ✓
History and Development
Epicyclic gearing is ancient (the ~2000-year-old Antikythera mechanism used it) and is the backbone of modern automatic transmissions, hub gears, and high-ratio compact reducers, prized for coaxial input/output and load sharing across multiple planets. Willis's 1841 formula and the tooth-count/assembly constraints are standard in Shigley and gear-design texts.
Related Concepts: Bevel Gear Pitch Angle, Gear Bending Stress Lewis, Worm Gear Efficiency, Gear Power Capacity, Gear Contact Stress
Notes: Computes the ring tooth-count constraint $Z_{ring}=Z_{sun}+2Z_{planet}$, not the speed ratio (which is $1+Z_{ring}/Z_{sun}$ with the ring fixed). Input symbols $R_s/R_p$ = sun/planet teeth; $R_c$ unused. Assembly needs $(Z_{sun}+Z_{ring})$ divisible by the planet count.