What Is the Planetary Epicyclic Gearbox Ratio & Torque Multiplier Calculator?
A planetary (epicyclic) gearbox is a coaxial gear transmission where planet gears orbit a central sun gear within an outer ring gear.
How Does the Planetary Epicyclic Gearbox Ratio & Torque Multiplier Calculator Work?
By fixing the ring gear, input into the sun gear rotates the carrier arm at a reduction ratio of 1 + (Z_ring / Z_sun) via Willis' relative velocity equation.
Planetary Epicyclic Gearbox Ratio & Torque Multiplier Calculator Formula & Variables
The core mathematical equation utilized by this calculator is expressed as:
i = 1 + \frac{Z_{\text{ring}}}{Z_{\text{sun}}}, \quad Z_{\text{planet}} = \frac{Z_{\text{ring}} - Z_{\text{sun}}}{2}, \quad T_{\text{out}} = T_{\text{in}} \cdot i \cdot \eta
Willis kinematic equation for fixed outer ring gear yields planetary reduction ratio i. Torque multiplies proportionally with mechanical efficiency.
How to Use the Planetary Epicyclic Gearbox Ratio & Torque Multiplier Calculator
- Enter tooth counts for the sun and ring gears.
- Select the drive configuration (which component is driven, fixed, and output).
- Input motor speed and torque to compute output mechanical performance.
Step-by-Step Example Calculation
Standard 4:1 Ring/Sun Planetary (Zs = 18, Zr = 72, Sun In / Carrier Out)
Input Values:
sunTeethZs:18
ringTeethZr:72
inputMember:sun-in-carrier-out
inputSpeedRpm:3000
inputTorqueNm:25.0
gearboxEfficiencyPercent:97.0
Worked Steps: Planet gears require 27 teeth. Gear reduction ratio is exactly 5.0:1, reducing 3,000 RPM down to 600 RPM while multiplying torque to 121.3 N·m.
Understanding Your Result
Gear Ratio: Speed reduction multiplier.
Planet Teeth: Required tooth count for each planet gear.
Output Torque: Multiplied usable shaft torque.
Factors That Affect the Result
- Tooth ratio governs reduction; tooth meshing friction and bearing drag dictate transmission efficiency.
When Should You Use This Calculator?
- Robotics actuator joints, electric vehicle wheel hubs, automatic transmissions, and aerospace turboprops.
Assumptions & Limitations
- Assumes spur gears with standard involute profile and symmetric planet carrier spacing.
Frequently Asked Questions
Calculation Accuracy & Reference Note
Standard Willis epicyclic kinematics formulation.