This simulation builds a line of two to four meshing spur gears and shows how tooth counts set the gear ratio. Each gear meshes with its neighbour, so the speed of one gear relative to the next follows ωout/ωin = −Nin/Nout. The minus sign means meshing gears spin in opposite directions, and the overall ratio is simply the product of the individual stage ratios along the train.
Sliders set the number of gears, each gear's tooth count (8–48 teeth), input speed (0–240 RPM) and input torque (1–50 N·m), while a checkbox makes gears B and C share a shaft to form a compound gear. The readout reports per-gear RPM, torque and direction, plus the overall ratio and mechanical advantage. Gear trains like these set the speed and torque of clocks, gearboxes, drills and bicycle drivetrains.
What is a gear train?
A gear train is a set of meshing gears that transmits rotation from an input shaft to an output shaft. By choosing different tooth counts, the train changes the output speed and torque relative to the input. This simulator lets you build a train of two to four spur gears and watch those effects in real time.
How is the gear ratio calculated?
For a single mesh the speed ratio equals the driven gear's tooth count divided by the driver's: ωout/ωin = −Nin/Nout. For a multi-stage train the overall ratio is the product of each stage. The simulator reports this as a reduction figure such as 2.500 : 1, meaning the output turns once for every 2.5 input turns.
Why do meshing gears spin in opposite directions?
Where two external gears mesh, their teeth move together at the contact point, so one surface moves up while the other moves down. That forces the two gears to rotate in opposite senses, which is why the ratio equation carries a minus sign. The arrows in the simulation show clockwise and counter-clockwise spins alternating along the train.
The first slider sets how many gears are in the train (2–4). A tooth-count slider for each gear sets its size from 8 to 48 teeth. The input speed slider sets the driver RPM (0–240) and the input torque slider sets the driving torque (1–50 N·m). Checkboxes toggle a compound gear and whether teeth are drawn, and buttons pause or reset the model.
Ignoring friction, power is conserved, so torque rises by exactly the same factor the speed falls. If the output turns at half the input speed, it delivers roughly twice the torque: Tout/Tin = Nout/Nin. The mechanical advantage shown in the readout is this same factor, which is why slow-turning gears can move heavy loads.
An idler is a gear placed between the driver and the final gear that meshes with both. It reverses the direction of rotation but does not change the overall gear ratio, because its tooth count cancels out of the maths. In a simple three- or four-gear train the simulator marks the middle gears as idlers to highlight this.
Ticking the compound box rigidly joins gears B and C on a shared shaft, so they turn at the same speed and direction. The train then has two separate meshes whose ratios multiply, allowing a much larger speed reduction in a compact space. This is how gearboxes achieve large ratios without using one enormous gear.
The kinematics are exact for ideal spur gears: speeds and torques follow the tooth-count ratios and direction reverses at each external mesh. It assumes constant module, no backlash and no friction or efficiency loss, so real gearboxes would deliver slightly less output torque. The drawn tooth shapes are simplified trapezoids rather than true involute profiles.
Module is the ratio of pitch diameter to tooth count, and meshing gears must share it so their teeth are the same size and spacing. In the simulator the pitch radius of each gear is drawn proportional to its tooth count, which keeps the pitch circles tangent at the contact point. Different tooth counts then change gear size while the teeth still mesh cleanly.
Mechanical advantage is how much the train multiplies torque, equal to the input speed divided by the output speed. A value of ×3 means the output torque is three times the input torque while turning a third as fast. The simulator displays it directly so you can see the trade-off between speed and force.
Gear trains appear in car gearboxes, electric-drill chucks, clocks and watches, bicycle hub gears, robotics and wind-turbine drivetrains. Engineers pick tooth counts to deliver the speed and torque a job needs, often combining several compound stages. Experimenting with the sliders here mirrors the choices a designer makes when sizing a real transmission.