🔬 The principle
What it means: a gear pair is two toothed wheels whose teeth lock together. Turn the driver and its teeth push the driven wheel's teeth, so the driven wheel must turn too. If the driven wheel has more teeth it turns slower — but each tooth pushes on a longer lever, so it pushes harder. In plain words: gears trade speed for torque (torque = turning force).
Why it matters: motors — and your legs — are happiest spinning fast with modest force. Wheels, robot arms and drills need slow, strong rotation. The gearbox is the translator between the two, and the gear ratio is the exchange rate it uses.
How it works: ① count the teeth: N₁ on the driver, N₂ on the driven wheel. ② At the mesh, teeth pass in lockstep — every tooth that leaves the driver is replaced by one arriving on the driven wheel — so the tooth-passing rate is equal on both: N₂·ω_out = N₁·ω_in, which gives ω_out = ω_in·N₁/N₂. ③ The contact force is equal and opposite on both wheels, but torque = force × radius and radius grows with tooth count, so torque scales the other way: τ_out = τ_in·N₂/N₁. ④ Power = torque × speed, and the two changes cancel — so power out can only equal power in minus losses; the difference leaves as friction heat at the mesh. ⑤ Because teeth push each other at the contact, the driven wheel always turns the opposite way to its driver.
Where you see it in real engineering
A car's gearbox keeps the engine near its efficient speed while the wheels need whatever the hill demands. Hobby robot motors spin ~9000 rpm with almost no usable torque, so every robot drive bolts a reduction gearbox onto them. Bicycles shift ratio with a derailleur; winches, conveyor drives, clockwork and robot-arm joints stack stages — and every extra stage in the chain costs a little more efficiency.
🧮 The maths
i = N₂/N₁, ω_out = ω_in / i, τ_out = τ_in·i·η, P_in = τ_in·ω_in·2π/60
N₁, N₂ — tooth counts, driver / driven (no unit) · i — gear ratio (no unit) · ω — angular speed (rpm; 1 rpm = 2π/60 rad/s) · τ — torque (N·m) · η — efficiency of ONE mesh, 0…1 (no unit) · P — power (W)
Two-stage train: i_total = (N₂/N₁)·(N₄/N₃) and η_total = η² — every mesh takes the tax again. P_out = η_total·P_in and the missing power is friction heat: P_loss = P_in − P_out (W).
Worked example: N₁ = 20, N₂ = 60 → i = 3. With ω_in = 90 rpm: ω_out = 90/3 = 30 rpm. With τ_in = 0.5 N·m and η = 0.95: τ_out = 0.5 × 3 × 0.95 = 1.43 N·m. Power in = 0.5 × 90 × 2π/60 = 4.71 W → power out = 0.95 × 4.71 = 4.48 W → 0.24 W of heat. Set those sliders — the tiles match.
Assumptions & limitations: teeth are drawn as simple rectangles (real involute profiles curve); η is a single lumped per-mesh number applied as a torque tax, and a two-stage train uses η²; no backlash, tooth bending, vibration or lubrication is modelled; speeds are steady — there is no spin-up inertia. This simulation is an educational approximation of a real spur-gear transmission: real gearboxes add bearing friction, oil churning and manufacturing tolerances, which this model folds into the single η slider.
🕹 Simulation workspace
Live measurements
Gear ratio i—
Output speed—
Output torque—
Transmission efficiency—
Power loss (heat)—
Input power—
Ready
Tiles are computed from the tooth counts and sliders every frame — nothing here is a fixed number. Two meshes in a row apply η twice (η²).
driver gear (input shaft)
stage-1 driven wheel
stage-2 driver (same shaft as N₂)
stage-2 output wheel
mesh contact point
marker spoke (watch the speeds differ)
📈 Graphs
Output speed vs gear ratio — the ideal curve ω_out = ω_in/i is a hyperbola: double the ratio, half the speed. The red dot is the current setting; in two-stage mode the x-axis is the TOTAL ratio. The view follows the dot.
Output torque vs gear ratio — a straight line τ_out = τ_in·i·η: torque is bought with ratio and taxed by η. Red dot = current setting (same x-axis as the left chart).
🌍 Real-world applications
🚗 Cars & EVsThe gearbox keeps the engine or motor near its efficient speed while the wheels get whatever the hill demands — first gear is a huge reduction for pulling away.
🤖 Robot drivetrainsHobby "TT" motors spin ~9000 rpm with almost no torque; the attached gearbox trades it down to ~200 rpm of useful push. Ratio first, motor second.
🚲 BicyclesDerailleur gears shift the ratio live: climbing wants a low gear (easy torque), sprinting a high gear (speed). Your legs are the input shaft.
🏭 IndustryWinches, conveyor drives, robot-arm joints and clockwork all stack stages — designers count the meshes because every one taxes efficiency.
🔧 Try it with hardware — Arduino UNO R4 WiFiPut an IR encoder wheel on a geared motor's output shaft, count pulses per second on a digital pin, and compare the measured rpm with your predicted ω_out = ω_in/i. The R4's LED matrix even makes a stroboscope.
⚙️ ESP32 + gearbox kitDrive a TT motor at several PWM duties, log output rpm each time, then divide by the printed gearbox ratio to estimate the bare motor speed — chain this sim with motor-torque like a real drivetrain.
✅ Check your understanding
📌 What you discovered
- The tooth ratio is the whole story: i = N₂/N₁ — speed divides by i, torque multiplies by i.
- Power is conserved minus heat: P_out = η·P_in — a gear train transforms power, it never creates it.
- Every mesh takes the η tax again: a two-stage train runs at η², so more stages = more heat.
- One reversal per mesh: a single stage turns the output backwards; two stages return it to the input's direction.
- Need speed up instead? Drive the big wheel: i < 1 is an overdrive pair and torque drops.
➡ Engineering takeaway: write down the ratio you need first, then count how many meshes you can afford — torque is bought with speed and paid for in heat.