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
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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