Why is Z_r forced to Z_s + 2 Z_p?

For standard **external** planet teeth meshing with a **sun** and an **internal** ring, the **center distance** from sun to planet is R_s + R_p and the ring’s **internal pitch radius** must be R_s + 2 R_p. With tooth count proportional to pitch radius on the same module, that gives **Z_r = Z_s + 2 Z_p** for three (or more) planets on the same orbit.

Can any Z_s and Z_p work with three planets in a real gearbox?

Not always. Even spacing needs extra **assembly** conditions on tooth counts (and sometimes hunting tooth). This demo always draws **three** equally spaced planets for clarity; treat that as a **pedagogical** picture when counts would not close in manufacturing.

What does negative RPM mean on the readouts?

It is **sign convention only**: opposite **sense of rotation** about the axis compared with the positive input. Magnitudes still follow the Willis equation for the chosen fixed member.

Where is torque ratio shown?

Only **kinematics** is simulated here. For an **ideal** lossless train, power is approximately **τ·ω** constant along the path, so torque ratios follow from the speed ratios—but friction and which member carries **reaction torque** matter in real hardware.

Planetary Gear Set

A planetary (epicyclic) gear train packs three main types of toothed members—sun, planets on a carrier, and an internal ring—into one coaxial stage. Holding one member and driving another yields different speed ratios in a compact package (common in automatic transmissions and hub reducers). This simulator uses the textbook Willis kinematic relation ω_s Z_s + ω_r Z_r = ω_c (Z_s + Z_r) for the three angular speeds (sun, ring, carrier) and enforces Z_r = Z_s + 2 Z_p so sun–planet–ring pitch radii stay consistent with three identical planets on one planet pitch circle. Planet spin about its own axis follows ω_p = ω_c − (Z_s/Z_p)(ω_s − ω_c) from meshing with the sun while moving with the carrier. You can hold the ring, the sun, or the carrier and drive the appropriate input with a signed RPM slider; readouts show all resulting speeds and the relevant ratio for that case. The animation is ideal: no tooth slip, friction, efficiency loss, or carrier balance conditions beyond the tooth-count rule—real designs must also satisfy assembly spacing (e.g. divisibility) for evenly spaced planets.

Who it's for: High school physics and introductory mechanical engineering students comparing serial gear trains to epicyclic stages; complements the existing Gear Train simulator.

Key terms

Planetary gear set
Epicyclic train
Willis equation
Sun gear
Ring gear (internal)
Planet carrier
Gear ratio
Hold member / brake

How it works

A planetary (epicyclic) gear set has a sun gear, a ring with internal teeth, and planets on a carrier. One member is often held while another is driven, giving compact speed reduction or overdrive in one stage. The fundamental kinematic constraint (Willis) is ω_s Z_s + ω_r Z_r = ω_c (Z_s + Z_r) with Z_r = Z_s + 2 Z_p for standard meshing. This demo uses three planets for a clear picture; real designs also require assembly spacing conditions on tooth counts.

Key equations

ω_s Z_s + ω_r Z_r = ω_c (Z_s + Z_r)

Z_r = Z_s + 2 Z_p (pitch geometry, standard set)

ω_p = ω_c − (Z_s/Z_p)(ω_s − ω_c) (planet spin)

Frequently asked questions

Why is Z_r forced to Z_s + 2 Z_p?: For standard external planet teeth meshing with a sun and an internal ring, the center distance from sun to planet is R_s + R_p and the ring’s internal pitch radius must be R_s + 2 R_p. With tooth count proportional to pitch radius on the same module, that gives Z_r = Z_s + 2 Z_p for three (or more) planets on the same orbit.
Can any Z_s and Z_p work with three planets in a real gearbox?: Not always. Even spacing needs extra assembly conditions on tooth counts (and sometimes hunting tooth). This demo always draws three equally spaced planets for clarity; treat that as a pedagogical picture when counts would not close in manufacturing.
What does negative RPM mean on the readouts?: It is sign convention only: opposite sense of rotation about the axis compared with the positive input. Magnitudes still follow the Willis equation for the chosen fixed member.
Where is torque ratio shown?: Only kinematics is simulated here. For an ideal lossless train, power is approximately τ·ω constant along the path, so torque ratios follow from the speed ratios—but friction and which member carries reaction torque matter in real hardware.

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