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
i = Z2 / Z1; n2 = n1 / i; T2 = T1 · i
Ratio i>1 reduces speed and increases torque.
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 notesRatio i = z2 / z1 = n1 / n2; driven speed n2 = n1 · z1 / z2. z = tooth count, n = speed.
Formula · Worked Example · Knowledge
Formula
The ratio of a single mesh is the ratio of driver/driven speeds or tooth counts:
$$i = \frac{Z_2}{Z_1} = \frac{N_\text{driver}}{N_\text{driven}}$$
$Z_1,Z_2$ 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:
$$i_\text{total}=\prod i_k=\frac{\prod Z_\text{driven}}{\prod Z_\text{driver}}$$
$$i = \frac{Z_2}{Z_1} = \frac{N_\text{driver}}{N_\text{driven}}$$
$Z_1,Z_2$ 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
In the train $Z_A=40,\ Z_{B1}=60,\ Z_{B2}=30,\ Z_{C1}=70,\ Z_{C2}=20,\ Z_D=80$. Find $i_{AD}$; if $n_A=1400\ \text{r/min}$, find $n_D$.
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:
$$n_D=\frac{1400}{14}=100\ \text{r/min}$$
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 $a=m(z_1+z_2)/2$.