Planetary Gearbox Ratio Calculation

Introduction

Planetary systems, or epicyclic gear systems, are essential in modern engineering, offering versatile speed variations. These systems are widely used in automatic car transmissions, industrial mixers, operating tables, and solar arrays. With four main componentsβ€”the ring gear, the sun gear, and the planetary gears connected to the carrierβ€”calculating the gear ratio may seem complex. However, the single-axis nature of these systems simplifies the process. This article outlines how to calculate planetary ratios effectively.

Gear Teeth Definitions

  • R: Number of teeth in the ring gear
  • S: Number of teeth in the sun gear
  • P: Number of teeth in the planet gears

Planetary Gearbox

Basic Constraints For planetary gears to function correctly, all teeth must have the same pitch (tooth spacing). The primary constraint is: 𝑅=2𝑃+𝑆 This means the number of teeth in the ring gear equals the number of teeth in the sun gear plus twice the number of teeth in the planet gears.

Example Calculation Consider a system with the following teeth counts:

  • Ring gear (R): 42 teeth
  • Sun gear (S): 18 teeth
  • Planet gears (P): 12 teeth

Using the constraint: 42=2Γ—12+18

Gear Ratio Calculation The gear ratio in planetary systems depends on the state of the carrier (Y), the sun gear (S), and the ring gear (R). The turns ratio formula is: (𝑅+𝑆)×𝑇𝑦=π‘…Γ—π‘‡π‘Ÿ+𝑆×𝑇𝑠

Three Key Scenarios

  1. Carrier as Input When the carrier is the input, and the sun gear is stationary: 𝑇𝑦=𝑇𝑠×𝑆𝑅+𝑆 Gear ratio: 𝑆𝑅+𝑆
  2. Carrier as Output When the carrier is the output, and the sun gear drives the system while the ring gear is stationary: 𝑇𝑦=𝑆𝑅 Gear ratio: 𝑆+𝑅𝑆
  3. Carrier Standing Still When the carrier is stationary, and the ring gear drives the system while the sun gear is the output: π‘‡π‘Ÿ=𝑅𝑆 Gear ratio: 𝑆𝑅

Additional Considerations

  • Even Spacing of Planet Gears For even spacing and synchronous engagement, the sum of the sun and ring gear teeth must be divisible by the number of planet gears. (𝑅+𝑆)mod  𝑁=0 where 𝑁 is the number of planet gears.
  • Uneven Spacing If uneven spacing is acceptable, this constraint is not required. However, the angle between planet gears is: Angle𝑝2𝑝=360𝑅+𝑆×𝑁

Planetary gear

Practical Applications

When using a planetary gear set with a stepper (e.g., NEMA 17) to increase torque, follow these steps:

  1. Calculate Teeth Numbers Determine the teeth on the sun and ring gears, then calculate the planet gears' teeth.
  2. Determine Gear Ratio Use the appropriate formula based on the carrier's state (input, output, or stationary).
  3. Apply the Gear Ratio For low-speed transmission with the carrier as output, sum the sun and ring gear teeth and divide by the drive gear teeth. For overdrive systems, divide the drive gear teeth by the sum of the sun and ring gear teeth.

Willis Equation for Planetary Gears

The fundamental equation derived by Willis describes the motion of the sun gear (S), ring gear (R), and carrier (C): π‘›π‘Ÿβ‹…π‘§π‘Ÿ=𝑛𝑐⋅(π‘§π‘Ÿ+𝑧𝑠)βˆ’π‘§π‘ β‹…π‘›π‘ 

Conclusion

Planetary gear systems are vital in engineering for their efficiency in speed variation. Calculating their gear ratios requires understanding the relationship between the teeth counts of the gears and the state of the carrier. By following the outlined steps and formulas, one can easily determine the gear ratios for various configurations, ensuring optimal performance in applications such as stepper motor-driven gearboxes.

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