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Gear Ratio Calculator

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Created by: Ethan Brooks

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This Gear Ratio Calculator helps you determine the gear ratio, output speed (RPM), and output torque (assuming 100% efficiency) from the number of teeth on the driving and driven gears. You can optionally provide input speed and torque for more detailed calculations.

What is a Gear Ratio Calculator?

A Gear Ratio Calculator is a tool used in mechanical engineering to determine the relationship between two or more intermeshing gears. The gear ratio is a fundamental concept that dictates how speed and torque are transmitted through a gear train.

This calculator helps you find the gear ratio given the number of teeth on the driving and driven gears. Optionally, if you provide input speed (RPM) or input torque, it can also calculate the corresponding output speed and output torque, assuming ideal conditions (100% efficiency).

Gear Ratio Formulas

Gear Ratio (GR)

GR = N₂ / N₁

Where:

  • GR = Gear Ratio
  • N₂ = Number of teeth on the driven (output) gear
  • N₁ = Number of teeth on the driving (input) gear

A gear ratio greater than 1 indicates a reduction in speed and an increase in torque (output gear is slower but more powerful). A gear ratio less than 1 indicates an increase in speed and a decrease in torque.

Output Speed (RPMout)

RPMout = RPMin / GR

Where:

  • RPMout = Rotational speed of the driven gear (in Revolutions Per Minute)
  • RPMin = Rotational speed of the driving gear (in Revolutions Per Minute)
  • GR = Gear Ratio

Output Torque (Τout)

Τout = Τin * GR * η

Where:

  • Τout = Torque of the driven gear (e.g., in Nm)
  • Τin = Torque of the driving gear (e.g., in Nm)
  • GR = Gear Ratio
  • η = Efficiency of the gear train (this calculator assumes η = 1 or 100% for simplicity)

For this calculator, we assume an ideal efficiency of 100% (η = 1), so the formula simplifies to: Τout = Τin * GR

How to Calculate Gear Ratio, Output Speed, and Torque: Example

Consider a simple gear train:

  • Number of teeth on driving gear (N₁) = 20 teeth
  • Number of teeth on driven gear (N₂) = 60 teeth
  • Input speed (RPMin) = 1200 RPM
  • Input torque (Τin) = 10 Nm
  1. Calculate Gear Ratio (GR):

    GR = N₂ / N₁ = 60 / 20 = 3

    (This is often expressed as 3:1)

  2. Calculate Output Speed (RPMout):

    RPMout = RPMin / GR = 1200 RPM / 3 = 400 RPM

  3. Calculate Output Torque (Τout) (assuming 100% efficiency):

    Τout = Τin * GR = 10 Nm * 3 = 30 Nm

So, the gear ratio is 3:1, the output speed is 400 RPM, and the output torque is 30 Nm.

Common Applications

  • Automotive Transmissions: Selecting appropriate gear ratios for different driving conditions (e.g., acceleration, cruising).
  • Robotics: Designing gearboxes for robot arms and actuators to achieve desired speed and torque.
  • Industrial Machinery: Used in conveyors, mixers, and other equipment to modify motor output.
  • Bicycles: Different gear combinations allow cyclists to adapt to varying terrains.
  • Clocks and Watches: Precise gear trains are essential for timekeeping mechanisms.

Sources and References

  1. Shigley, J. E., Mischke, C. R., & Budynas, R. G. (2004). *Mechanical Engineering Design* (7th ed.). McGraw-Hill.
  2. Norton, R. L. (2008). *Machine Design: An Integrated Approach* (3rd ed.). Pearson Prentice Hall.
  3. Spotts, M. F., Shoup, T. E., & Hornberger, L. E. (2004). *Design of Machine Elements* (8th ed.). Pearson Prentice Hall.