Calculating the output torque in a geared system relies on understanding the gear ratio and the fundamental principles of mechanical advantage.
How to Calculate Gear Ratio Torque (Output Torque)
While "gear ratio torque" isn't a specific value, the gear ratio is crucial for calculating the output torque generated by a gearbox or gear train when you know the input torque. Essentially, a gear ratio determines how much the input torque is multiplied (or divided) at the output shaft.
The relationship between gear ratio, input torque, and output torque in an ideal (lossless) system is:
Output Torque = Input Torque × Gear Ratio
This means if you have a gear ratio greater than 1, the output torque will be higher than the input torque, but the output speed will be lower. If the gear ratio is less than 1, the output torque will be lower than the input torque, and the output speed will be higher.
Understanding the Gear Ratio
The gear ratio itself is a measure of the mechanical advantage provided by the gear system. As noted in the reference:
The gear ratio of a gearbox is usually expressed in turns of the input to turns of the output. To find this you divide the number of teeth on the bull gear (the gear on the output side) by the number of teeth on the pinion gear (the gear on the input side).
So, the calculation for the gear ratio is:
Gear Ratio (GR) = Number of Teeth on Output Gear / Number of Teeth on Input Gear
Also commonly expressed as:
GR = T_out / T_in
Where:
T_out= Number of teeth on the output gear (Bull Gear)T_in= Number of teeth on the input gear (Pinion Gear)
This ratio also inversely relates the input and output speeds:
Output Speed = Input Speed / Gear Ratio
The Torque-Speed Relationship
The inverse relationship between speed and torque in a gear system is based on the principle of conservation of power (ideally, ignoring friction and other losses). Power is proportional to Torque × Speed. If power is conserved, then:
Input Torque × Input Speed ≈ Output Torque × Output Speed
Rearranging this equation shows how the gear ratio influences torque:
Output Torque / Input Torque ≈ Input Speed / Output Speed
Since Input Speed / Output Speed is equal to the Gear Ratio (derived from Output Speed = Input Speed / Gear Ratio), we get:
Output Torque / Input Torque ≈ Gear Ratio
Which leads back to the primary formula:
Output Torque ≈ Input Torque × Gear Ratio
Factors Affecting Real-World Torque Output
While the ideal calculation provides a theoretical maximum output torque, real-world systems always have losses due to friction, heat, and other inefficiencies. Therefore, the actual output torque will be less than the ideal calculated value.
To account for this, an efficiency factor is used:
Actual Output Torque = Input Torque × Gear Ratio × Efficiency (η)
Efficiency (η) is typically a value between 0 and 1 (or a percentage between 0% and 100%).
Example Calculation
Let's say you have a gear system with:
- Input gear (Pinion) with 20 teeth
- Output gear (Bull Gear) with 80 teeth
- Input Torque = 50 Nm
- Efficiency = 90% (or 0.9)
-
Calculate the Gear Ratio:
GR = (Teeth on Output Gear) / (Teeth on Input Gear)
GR = 80 / 20
GR = 4 -
Calculate the Ideal Output Torque:
Ideal Output Torque = Input Torque × Gear Ratio
Ideal Output Torque = 50 Nm × 4
Ideal Output Torque = 200 Nm -
Calculate the Actual Output Torque (considering efficiency):
Actual Output Torque = Ideal Output Torque × Efficiency
Actual Output Torque = 200 Nm × 0.9
Actual Output Torque = 180 Nm
Summary Table
| Metric | Formula | Notes |
|---|---|---|
| Gear Ratio (GR) | T_out / T_in |
T_out = Output Teeth, T_in = Input Teeth |
| Ideal Output Torque | Input Torque * GR |
Assumes 100% efficiency |
| Actual Output Torque | Input Torque * GR * Efficiency (η) |
Accounts for system losses |
Understanding the gear ratio derived from the number of teeth allows you to predict and calculate the corresponding ideal (and actual, with efficiency) output torque for a given input torque.
