Formula & Calculator
Gear Ratio
Calculates the speed reduction or increase produced by a pair of meshing gears based on their number of teeth.
Interpretation
Gear ratio is the ratio of the number of teeth on the driven gear to that on the driver gear. It determines the speed and torque multiplication: GR = N_driven / N_driver. Higher ratios reduce speed but increase torque.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| GR | Gear ratio (dimensionless) | |
| N_driven | Number of teeth on the driven gear | |
| N_driver | Number of teeth on the driver (input) gear |
What it means
Gear ratio (GR) is defined as the ratio of the number of teeth on the driven gear to the number of teeth on the driver gear: GR = Z_driven / Z_driver. Since the meshing gears have the same module, the ratio also equals the pitch diameters. The gear ratio determines the speed reduction or increase: output speed = input speed / GR. It also determines torque multiplication: output torque = input torque × GR (assuming no losses). This is the basis for gearboxes in vehicles, where different gears provide varying ratios to match engine speed to driving conditions. In industrial machinery, gear trains are used to adjust speed and torque for specific applications. The number of teeth must be integers, leading to fixed discrete ratios. Gear trains can be compounded to achieve large ratios. The efficiency of gear drives is high (typically 95‑98%). Understanding gear ratios is essential for designing power transmission systems, robotic joints, and instrumentation. It also affects the inertia reflected to the motor.
Worked example
Gear Ratio – Two Examples
Real‑World| Parameter | Value |
|---|---|
| N_driver | 20 |
| N_driven | 60 |
| Parameter | Value |
|---|---|
| N_driver | 40 |
| N_driven | 20 |
Common mistakes
- Teeth count: Use the number of teeth, not the diameter (though they are proportional).
- Driver vs. driven: GR = Ndriven / Ndriver; a GR > 1 means speed reduction, torque increase.
- Speed and torque: Speed ratio is inverse of torque ratio (neglecting losses).
- Compound gears: For multiple gear pairs, multiply individual gear ratios.
- Direction: External gears reverse rotation; internal gears maintain direction.
Applications
Gear ratio is the ratio of teeth on the driven gear to that on the driver gear, determining speed and torque multiplication. Gears are ubiquitous in mechanical power transmission, from automotive transmissions to industrial gearboxes. By selecting the appropriate gear ratio, engineers can achieve the desired balance between speed and torque for a given application. In vehicles, gear ratios allow engines to operate efficiently over a wide range of speeds. In robotics, gears are used to amplify motor torque. The gear ratio also affects efficiency, noise, and wear. Understanding gear ratios is essential for designing reliable and efficient mechanical systems that convert input power into useful output.
- Automotive transmission and differential design
- Industrial gearbox and speed reducer design
- Robotic joint and actuator torque amplification
- Clock and watch mechanism design
- Wind turbine and marine propulsion gearing
Frequently Asked Questions
The gear ratio is the ratio of the number of teeth on the driven gear to the number of teeth on the driver gear: GR = N_driven / N_driver. It determines how the rotational speed and torque are transformed. If GR > 1, the output turns slower but with higher torque (speed reduction).
For an ideal (lossless) gear pair:
- Speed: ω_driven / ω_driver = N_driver / N_driven = 1/GR.
- Torque: T_driven / T_driver = N_driven / N_driver = GR.
- Inverting the ratio – if you use N_driver/N_driven, you get the inverse gear ratio. Always define GR = N_driven/N_driver for speed reduction.
- Using diameters instead of teeth – gear ratio is based on tooth count; using pitch diameters is equivalent only if they have the same module.
- Ignoring the gear train with multiple meshes – for a train, the overall ratio is the product of the individual ratios.
- Confusing driver and driven – always identify which gear is the input (driver) and which is the output (driven).
For a compound train (multiple meshes), the overall ratio is the product of the ratios of each meshing pair: GR_total = (N₂/N₁) × (N₄/N₃) × ..., where each pair is a driver‑driven mesh. Alternatively, GR_total = (product of driven teeth) / (product of driver teeth).
If the driven gear has more teeth than the driver (N_driven > N_driver), it is a speed reducer (output slower, higher torque). If the driven has fewer teeth, it is a speed increaser (output faster, lower torque).
The mechanical advantage (MA) is the ratio of output torque to input torque, which equals the gear ratio (ignoring losses). A gearbox with a GR of 4 multiplies the input torque by 4, but reduces the output speed by a factor of 4.
You need to know the required output speed and torque. Given the input speed and torque of the motor, choose GR = (input speed) / (desired output speed) = (desired output torque) / (input torque). Then select gears with that tooth ratio.
In a manual transmission, the transmission ratio is the overall ratio from the input shaft to the output shaft, which includes the gear pair, the final drive ratio, and any other gearing. The gear ratio usually refers to a single gear mesh.
Each gear mesh has an efficiency (η ≈ 98‑99%). For a high ratio, multiple meshes are often needed, and the overall efficiency decreases: η_total = η₁ × η₂ × ... . High reduction ratios (e.g., > 10) often use worm gears or planetary gears to maintain efficiency.
The output angular acceleration is related to the input by α_output = α_input / GR (for a reducer). Additionally, the inertia reflected to the input is J_reflected = J_output × (1/GR²). This is important in dynamic analysis of servo systems.