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Experiment 008 / Robotics and control

Gear ratio

Calculate speed, ratio and direction for an external gear pair or an internal ring gear.

Workbench / Explicit model ● In your browser
01 / Adjust the model
teeth
teeth
rpm
Try an example

02 / Transmit motion

s
Output speed −30 rpm
Ratio Z2/Z1 2
Relative direction Opposite
Input rotation 0°
Output rotation 0°
Model time 0 s

The output gear has twice as many teeth: it turns at half the speed in the opposite direction.

Positive means counterclockwise when viewed from the front. Teeth are schematic. Animation limits visible input speed to 10 rpm so motion is easy to follow; the numbers show model speeds.

Use it in your classroom

Share this exercise with its current values. The link contains only numbers and model options; do not add names or email addresses.

The code includes a text link to the full explanation outside the frame. Animation starts paused. Adjust the iframe height if your platform needs it.

Two input revolutions, one output revolution

A 20-tooth gear drives a 40-tooth gear. With an input of 60 rpm in the positive direction, the output is −30 rpm. Z2/Z1 = 2: two input revolutions produce one output revolution. The negative sign indicates opposite directions.

Swap the tooth counts: 40 input teeth and 20 output teeth double the speed. With an internal ring gear, contact changes sides and both gears turn in the same direction.

Tangential speeds must match

Ratio i = Z2 / Z1
External mesh: n2 = −n1 × Z1/Z2
Internal mesh: n2 = +n1 × Z1/Z2
Accumulated rotation θ = n × 360° × t/60 = 6nt

Enter n in revolutions per minute and t in seconds. Compatible gears have pitch diameters proportional to tooth count. Both must travel the same distance at contact, giving the inverse tooth-count-to-speed ratio.

Internal mode treats Z2 as the ring gear and Z1 as the pinion, so Z2 must exceed Z1. Absolute signs follow the counterclockwise-positive convention; a negative input reverses both motions.

This calculates motion, not manufacturable gears

Only one fixed-axis gear pair with ideal contact is modeled. There is no slip, backlash, elasticity or loss. The drawn profiles are not involutes and cannot be used to manufacture gears. The tool permits 4–120 teeth, but real interference or geometry would make some pairs impractical.

Forces, gear module, mounting distance, efficiency and torque are not calculated. This is not a planetary train: the output ring rotates about a fixed axis with no moving carrier.

Common mistakes

  • Multiplying by Z2/Z1 when finding output speed: that is the reduction ratio, not the speed factor.
  • Forgetting the sign when comparing rotation directions.
  • Confusing the deliberately slowed animation with the calculated rpm.

Try it yourself

12 teeth drive 36 at 120 rpm. What is the external output speed?

−120 × 12/36 = −40 rpm. The reduction ratio is 3.

A 20-tooth pinion drives a 60-tooth ring with a −90 rpm input. What happens?

The output is −30 rpm: both gears rotate clockwise, and the ring turns three times more slowly.