M87GearLab | Planetary Gear Fundamentals

01 Anatomy

What makes a planetary gearset different?

A planetary gearset puts several meshes into one compact package. The planets mesh with the sun gear and with an internal ring gear, while the carrier holds the planet shafts and may also rotate as an output or input member.

Sun gear

The central external gear. It often serves as the high-speed member because it is small and concentric with the output axis.

Planet gears and carrier

The planets spin on their own axes while orbiting with the carrier. Multiple planets share load, but only when spacing, stiffness, and manufacturing are handled well.

Internal ring

The outer internal gear. Its teeth face inward, so the same module and compatible pressure angle must be used for the planet-ring mesh.

02 Tooth Counts

The first relationship is simple: the ring must fit the sun and planets.

In a simple planetary set with equal module and no exotic spacing trick, the ring tooth count is tied to the sun and planet tooth counts.

Ring tooth count

R = S + 2P

S is sun teeth, P is planet teeth, and R is ring teeth. The planet sits between the sun pitch circle and the ring pitch circle, so the ring has to account for the sun plus two planet pitch radii.

Pitch radius and center distance

d = m z a_sw = m (S + P) / 2

m is module, z is tooth count, and a_sw is sun-to-planet center distance. The same center also keeps the planet tangent to the internal ring pitch circle.

03 Standard Geometry

Module, pressure angle, backlash, and clearance set the working envelope.

The geometry concepts below are standard gear vocabulary. They describe the envelope a gear designer reasons about when checking fit, clearance, and contact behavior.

Concept	Practical meaning	Common formula or note
Module	Metric tooth size. Larger module means larger teeth for the same tooth count.	`m = d / z`
Pitch diameter	The reference diameter where ideal rolling contact is measured.	`d = m z`
Pressure angle	The angle of force transfer along the line of action. Many standard spur gears use values around 20 degrees.	`d_b = d cos(alpha)` for base diameter
Backlash	Small intentional clearance between mating teeth so the mesh can move after manufacturing variation, thermal changes, and lubrication.	Too little can bind; too much can feel loose or noisy.
Clearance	Space between a tooth tip and the opposite root. It keeps tips from bottoming out.	Standard full-depth proportions often start near `c* = 0.25`.

04 Motion Math

Planetary ratios are easiest when measured relative to the carrier.

The classic relationship is often called Willis' equation. It compares the sun and ring speeds after subtracting the carrier speed from both.

Carrier-Relative View

Subtract carrier speed from the moving members. The remaining sun/ring relationship behaves like a fixed-axis internal gear pair.

Carrier frame transform Freeze the carrier, then compare the sun and ring.

omega_c -> 0

1

Shift every speed into the carrier frame. member speed - omega_c

2

Write the two relative speeds. omega_s' = omega_s - omega_c omega_r' = omega_r - omega_c

3

Apply the internal gear ratio. omega_s' / omega_r' = -R / S

Willis relationship (omega_s - omega_c) / (omega_r - omega_c) = -R / S

Willis equation

(omega_s - omega_c) / (omega_r - omega_c) = -R / S

omega_s, omega_r, and omega_c are the angular speeds of the sun, ring, and carrier. The minus sign comes from the internal gear relationship.

Fixed-ring reduction

ratio = omega_s / omega_c = 1 + R / S

If the ring is fixed and the sun drives the set, the carrier turns slower than the sun. This is a common compact reduction layout.

05 Indexing

Even planet spacing is not just visual symmetry.

If the planets are evenly spaced, the teeth also need to arrive at compatible mesh phases. Otherwise the drawing may look circular, but the assembly can be impossible or uneven.

Even planet indexing

(S + R) / N = integer

N is the number of planets. Since R = S + 2P, this is equivalent to checking whether 2S + 2P divides cleanly by N.

Spacing clearance

Indexing tells you whether the tooth phases can line up. It does not prove the planet bodies have enough space between them, that shafts are strong enough, or that tolerance stackups are acceptable.

06 Hunting Tooth

Wear improves when the same teeth do not keep meeting too often.

A hunting-tooth relationship distributes contact across more tooth pairs over time. The simple way to reason about this is with common divisors and repeat periods.

Repeat period

mesh_repeat = lcm(a, b) shared_factor = gcd(a, b)

For two mating gears with tooth counts a and b, a low greatest common divisor generally means better distribution of tooth contacts.

Planetary sets need both meshes

A planetary design should consider sun-planet and planet-ring contact behavior. A strong-looking ratio can still repeat contact patterns more often than desired.

07 Engineering Reality

Good math is necessary, but it is not the whole gearbox.

Gear design lives between geometry, material behavior, manufacturing process, lubrication, tolerance stackup, shaft stiffness, housing stiffness, speed, temperature, duty cycle, and abuse cases.

Load and material estimates

Load indicators help catch obvious risk, compare material presets, and guide early decisions. They do not replace laboratory testing or a formal strength rating.

Validation checks

Validation can flag practical geometry concerns such as spacing, contact, wall thickness, bore margin, and assembly conflicts. It should be treated as design review assistance.

Manufacturing handoff

Exports should be inspected in the target CAD, CAM, or slicer workflow. Always confirm units, scale, tolerances, clearances, and tool/process limitations before production.