Interactive Epicyclic Gear Visualization

This interactive tool shows planetary gears in real time and lets you change geometry and speeds while the animation runs. Planetary gears rotate and mesh automatically on screen and all kinematic values update instantly. Adjust tooth counts, module and input speed to explore behavior across a wide range of designs.

What a planetary gear set is

1. Sun gear located at the center
2. Ring gear that is an outer ring with internal teeth
3. Planet gears that orbit the sun and roll along the ring while mounted on a carrier

These assemblies are common in automatic transmissions, bicycle hub gears, electric drive reducers and precision clockwork. Use the simulator to test tooth counts, confirm mesh conditions and preview motion before detailed design or prototyping.

Key parameters, formulas and worked examples

Mesh condition that prevents interference is the tooth count rule

Zr = Zs + 2 × Zp

When Zs and Zp are chosen this formula gives the required Zr so planetary teeth align and planets fit evenly around the sun.

Pitch circle diameter for any gear follows from module multiplied by tooth count

D = m × Z

Center to center distance between sun and planet equals half the sum of their pitch diameters

Asp = Ds + Dp divided by 2

Center distance between planet and ring equals half the difference between ring and planet pitch diameters

Apr = Dr minus Dp divided by 2

For correct geometry Asp and Apr must be equal

If the ring is fixed the simple ratio for carrier speed relative to sun speed is

i = 1 + Zr divided by Zs

More generally the Willis relation links all three angular speeds

n_s minus n_c divided by n_r minus n_c equals minus Zr divided by Zs

Here n_s is sun speed, n_r is ring speed and n_c is carrier speed. Substitute zero for any fixed member to simplify calculations.

Worked example with different numbers

• Zs equals 18
• Zp equals 26
• Zr equals 70 because Zr equals Zs plus 2 times Zp
• Sun rotates at 120 RPM
• Ring is held stationary

i = 1 + 70 divided by 18 equals 4.8889 approximately

Carrier speed equals sun speed divided by i equals 120 divided by 4.8889 equals 24.55 RPM approximately

Result shows the carrier turns about five times slower than the sun in this arrangement. Use the simulator to vary Zp or Zs and observe how the ratio and carrier motion change.

Suggested tooth sets for common modules

Where planetary gears are used

• Automotive transmissions and hybrid drivetrains
• Bicycle hub gear systems
• Compact reducers for robotics and industrial drives
• Precision timekeeping and watchwork
• Power tools where compact high reduction is required

Practical design and verification tips

• Always verify Zr equals Zs plus 2 times Zp before detailed layout
• Use module and tooth counts that give integer center distances in millimeters to simplify bearing and carrier design
• Check for undercut and interference at low tooth counts and increase module or tooth count as necessary
• For power transmission estimate load sharing between planets and size bearings accordingly
• When testing with the simulator capture values in pitch diameter and speeds for CAD input and validation

Conclusion: The simulator speeds up geometry checks and kinematic verification for planetary gear systems. It helps avoid common mistakes in tooth selection and center distances and is useful for teaching, initial design and quick iteration.

Recommended reading

• Shigley Mechanical Engineering Design
• AGMA Fundamentals of Gear Design
• Design of Machine Elements by V B Bhandari

David Parry — Senior Engineering Analyst

Specializing in electronics and physics-based simulations with 20+ years of engineering experience. David ensures the mathematical and physical accuracy of the tools at ProCalcLab.