Gear Ratio

Calculate gear ratio, output speed and torque for gear pairs.

Inputs

Z_1

Driver

Z_2

Driven

Input Speed (RPM)(optional)

RPM

Input Torque (N·m)(optional)

N·m

Formula Interpretation

Single Pair Gear Ratio

i = \dfrac{Z_2}{Z_1} = \dfrac{N_\text{driver}}{N_\text{driven}}

$i$ is the gear ratio; $Z_2$ is driven gear teeth, $Z_1$ is driver gear teeth. More teeth = lower speed. $i$ > 1 means speed reduction.

Output Speed

n_\text{out} = \dfrac{n_\text{in}}{i}

$n_{\text{out}}$ is the output shaft speed (RPM); $n_{\text{in}}$ is the input speed. Divide by gear ratio $i$ . Higher ratio = lower output speed.

Output Torque

T_\text{out} = T_\text{in} \times i

$T_{\text{out}}$ is the output torque (N·m); $T_{\text{in}}$ is the input torque. Multiply by $i$ (friction losses ignored). Speed reduction amplifies torque proportionally.

Gear Train Ratio (Multi-stage)

i_{AD} = \dfrac{Z_{B1} \cdot Z_{C1} \cdot Z_D}{Z_A \cdot Z_{B2} \cdot Z_{C2}}

Overall ratio = product of all driven teeth ÷ product of all driver teeth. Multiplying stage ratios achieves large speed ratios in compact space.

Output Speed of Gear Train

N_D = \dfrac{N_A}{i_{AD}}

$N_D$ is the output speed at shaft D; $N_A$ is the input speed at shaft A (RPM). Divide $N_A$ by total ratio $i_{AD}$ to get $N_D$ .

Knowledge: Gear Trains

A gear train is a combination of multiple gear pairs designed to achieve a required speed ratio and direction. When a large speed ratio must be achieved in a small space, a gear train is essential. The overall ratio of a gear train equals the product of all the individual stage ratios — or equivalently, the product of all driven gear tooth counts divided by the product of all driver gear tooth counts.

Example

In the gear train shown, $Z_A = 40$ , $Z_{B1} = 60$ , $Z_{B2} = 30$ , $Z_{C1} = 70$ , $Z_{C2} = 20$ , $Z_D = 80$ . Find the overall ratio $i_{AD}$ . If the input speed at A is $1400\,\text{r/min}$ , what is the output speed at D?

Step 1 — Calculate ratio $i_{AD}$

i_{AD} = \dfrac{Z_{B1} \cdot Z_{C1} \cdot Z_D}{Z_A \cdot Z_{B2} \cdot Z_{C2}} = \dfrac{60 \times 70 \times 80}{40 \times 30 \times 20} = \dfrac{336000}{24000} = 14

Step 2 — Calculate output speed

N_D = \dfrac{N_A}{i_{AD}} = \dfrac{1400}{14} = 100 \text{ r/min}

Therefore, the output speed at shaft D is $N_D = 100\,\text{r/min}$ .

Extended Knowledge

•In a simple external gear train with three gears, the middle gear (idler gear) does not affect the overall speed ratio — it only changes the direction of rotation.
•When calculating gear train ratios, always carefully distinguish between driver gears and driven gears at each meshing stage.

Related Standards & Articles

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Gear Ratio

Calculate gear ratio, output speed and torque for gear pairs.

Inputs

Z_1

Driver

Z_2

Driven

Input Speed (RPM)(optional)

RPM

Input Torque (N·m)(optional)

N·m

Formula Interpretation

Single Pair Gear Ratio

i = \dfrac{Z_2}{Z_1} = \dfrac{N_\text{driver}}{N_\text{driven}}

$i$ is the gear ratio; $Z_2$ is driven gear teeth, $Z_1$ is driver gear teeth. More teeth = lower speed. $i$ > 1 means speed reduction.

Output Speed

n_\text{out} = \dfrac{n_\text{in}}{i}

$n_{\text{out}}$ is the output shaft speed (RPM); $n_{\text{in}}$ is the input speed. Divide by gear ratio $i$ . Higher ratio = lower output speed.

Output Torque

T_\text{out} = T_\text{in} \times i

$T_{\text{out}}$ is the output torque (N·m); $T_{\text{in}}$ is the input torque. Multiply by $i$ (friction losses ignored). Speed reduction amplifies torque proportionally.

Gear Train Ratio (Multi-stage)

i_{AD} = \dfrac{Z_{B1} \cdot Z_{C1} \cdot Z_D}{Z_A \cdot Z_{B2} \cdot Z_{C2}}

Overall ratio = product of all driven teeth ÷ product of all driver teeth. Multiplying stage ratios achieves large speed ratios in compact space.

Output Speed of Gear Train

N_D = \dfrac{N_A}{i_{AD}}

$N_D$ is the output speed at shaft D; $N_A$ is the input speed at shaft A (RPM). Divide $N_A$ by total ratio $i_{AD}$ to get $N_D$ .

Knowledge: Gear Trains

Example

Step 1 — Calculate ratio $i_{AD}$

i_{AD} = \dfrac{Z_{B1} \cdot Z_{C1} \cdot Z_D}{Z_A \cdot Z_{B2} \cdot Z_{C2}} = \dfrac{60 \times 70 \times 80}{40 \times 30 \times 20} = \dfrac{336000}{24000} = 14

Step 2 — Calculate output speed

N_D = \dfrac{N_A}{i_{AD}} = \dfrac{1400}{14} = 100 \text{ r/min}

Therefore, the output speed at shaft D is $N_D = 100\,\text{r/min}$ .

Extended Knowledge

•In a simple external gear train with three gears, the middle gear (idler gear) does not affect the overall speed ratio — it only changes the direction of rotation.
•When calculating gear train ratios, always carefully distinguish between driver gears and driven gears at each meshing stage.

Related Standards & Articles

Authoritative references for the formulas used in this calculator

Article

Gear Design & Selection Guide: From Parameters to System Design

Standard

Gear Design Symbol Reference (GB/T 2821 / ISO 701)

Standard