Gear Ratio
Compute the gear ratio between driver and driven gears, output shaft speed (RPM) and output torque for spur, helical and bevel pairs (AGMA).
Inputs
Formula
Fundamentals
Gear ratio describes the relative speed (or tooth count) of driver vs driven gear — a core parameter in transmission design.
For a meshing pair, $i=Z_2/Z_1=n_1/n_2$. $i>1$ is reduction (lower speed, higher torque); $i<1$ overdrive. A multi-stage train's overall ratio equals the product of stage ratios.
History
Gears trace back to ancient Greece (Aristotle) and Rome (Vitruvius' water mills); Han-dynasty China used them in chain pumps. Modern involute gearing was established by Euler and Grant in the 18th c., and standard modules enabled interchangeability in 1908.
Engineering applications
Used in car transmissions, industrial reducers, clocks, robot joints and wind-turbine speed-up gearboxes to match speed and torque precisely.
Glossary
| Ratio $i$ | Driven speed / driver speed, also driver teeth / driven teeth. |
| Module $m$ | Basic tooth-size parameter; only equal modules mesh correctly. |
| Center distance $a$ | Distance between gear axes, $a=m(z_1+z_2)/2$. |
How to use
- Fill in Driver teeth Z₁, Driven teeth Z₂, Input speed n₁, Input torque T₁ in the Inputs section (watch the unit on each field).
- Click Calculate; the tool evaluates the formula shown above.
- Read Ratio i, Output speed, Output torque in the results area.
Formula · Worked Example · Knowledge
Formula
$$i = \frac{Z_2}{Z_1} = \frac{N_\text{driver}}{N_\text{driven}}$$
$Z_1,Z_2$ are driver/driven tooth counts, $N$ is speed (r/min). Output speed $n_\text{out}=n_\text{in}/i$. $i>1$ means reduction, $i<1$ overdrive.
A multi-stage gear train's overall ratio equals the product of stage ratios, also the product of all driven teeth over all driver teeth:
$$i_\text{total}=\prod i_k=\frac{\prod Z_\text{driven}}{\prod Z_\text{driver}}$$
Worked Example
Step 1 — overall ratio (drivers A,B2,C2; driven B1,C1,D):
$$i_{AD}=\frac{60\times70\times80}{40\times30\times20}=14$$
Step 2 — speed of D:
$$n_D=\frac{1400}{14}=100\ \text{r/min}$$
Key Points
- A gear train combines several meshes to achieve a desired direction and large ratio in limited space.
- Overall ratio = product of stage ratios = product of driven teeth / product of driver teeth.
- More teeth → lower speed. Only gears with the same module $m$ mesh; center distance $a=m(z_1+z_2)/2$.
Parameters
Inputs
| Parameter | Symbol | Unit | Default |
|---|---|---|---|
| Driver teeth Z₁ | Z1 | 20 | |
| Driven teeth Z₂ | Z2 | 40 | |
| Input speed n₁ (rpm) | n1 | rpm | 1500 |
| Input torque T₁ (N·m) | T1 | N·m | 10 |
Outputs
| Result | Symbol | Unit |
|---|---|---|
| Ratio i | i | |
| Output speed | n2 | rpm |
| Output torque | T2 | N·m |
Applications
- Common engineering use cases